If you read the other answers, it should be apparent that the qualities you seek such as (a) better portraits and (b) the desire to have a blurred background ... aren't really one thing, but a combination of many factors.
There are some nuances but the short answer is ... portraits do not require advanced DSLRs (so entry level is fine) but... there are nuances to consider. I discuss these below.
The advantage of the DSLR isn't that the camera is 'better' per se, but rather that it allows interchangeable accessories (such as lenses, lighting, etc.) to create the right conditions in order to capture the results you want.
Much of this is based on the knowledge & skill of the photographer. Buying a more expensive musical instrument doesn't make a person a better musician... learning music and practicing makes a person a better musician. The camera can't propel you forward ... but a camera with limited capabilities might hold you back.
If I buy a better guitar than anything Peter Frampton uses ... I will still not be a better musician than Peter Frampton (nowhere even close). His knowledge and skill in that area of music is legendary and mine ... not so much.
We're getting into a subjective area, but flip through portraits you like and try to notice why you like them. Inspect the posing of the model, the composition of the frame and how foreground or background elements are used. Check out the lighting (especially check out the lighting).
My personal thoughts are that:
- Photographer knowledge & experience (skill) is probably the most important factor to influence the results. There's no getting around the notion that there will be a lot of learning and a lot of practice. The camera itself isn't a substitute for those considerations ... no matter how much a person pays for the equipment. Compositional skills, posing skills, exposure skills, lighting skills, etc. all require knowledge & experience ... regardless of how good the gear is. The camera will offer an 'automatic' mode and while that mode will capture adequate shots, they probably will not be the artistic results you were hoping to get.
- Lighting is next. I put this ahead of lens selection. A key idea here is that you cannot have good light ... without good shadow. An image of a full moon never looks as good as an image of a 1st Quarter moon. The difference is that one has 'flat' lighting and the other has the light coming from the side. When the light comes from the side, any 3D textures produce shadows. It's the interplay of those highlights and shadows that causes the subject to appear three-dimensional with loads of textures. This isn't just true of the moon... it's true of anything you photograph. Shadows queue the eye (and the human brain) and provide information about the textures and contours. Another consideration is whether the transition from highlight to shadow is an abrupt line ... or a gentle transition. Is it 'hard' lighting or 'soft' lighting (hard & soft refer to those transitions... a hard-edge has an abrupt transition with a sharp line separating light & shadow. A soft-edge is a very gentle transition from one to the other.) Lighting can create moods... you can use light to convey emotions such as joy or peace or despair. In my opinion this is one of the most powerful tools a photographer has (and yet so very many photographers seem to be obsessed with just their camera.) The best lighting is the lighting that does what you want ... and this often means you may need ways to control the lighting (which is why advanced photographers own auxiliary lighting and lighting modifiers).
- Lens selection is next on my personal list of priorities. A helpful way to think of lenses isn't so much by focal length ... but by angle-of-view. A lens can be wide, normal, or narrow. A normal angle of view is one that matches roughly what a human eye would perceive. A technical point to keep in mind (it's not hard to remember this) is that a lens will provide a normal angle of view if the focal length of the lens is the same as the diagonal measure of the sensor. For most DSLRs with an APS-C size sensor this is roughly 28mm (that's not exact). A 28mm lens on such a camera will offer a normal angle of view. If you use a shorter focal length (e.g. 20mm) you will have a moderately wide angle of view. If you use a much lower focal length (e.g. 10mm) you will have a very wide angle of view. A 50mm focal length will offer a moderately narrow angle of view. A 200mm lens will offer a very narrow angle of view. These angles-of-view create some interesting and useful side-effects. Wide angle lenses do not just shoot wider scenes... they also stretch the sense of the depth in a scene (want to make a room look larger or make a subject appear farther away... use a wider lens). The opposite happens with a narrow lens. Narrow lenses (long focal lengths) produce compression. Far away subjects don't look so far. The sense of depth in a scene is 'compressed'.
- The camera body is in last place on my list. It isn't that it is not important... it is important. But it wields less influence than the three factors above it. There are many instances where the physical sensor used in an entry level body is actually the same sensor used in a higher-end body. So what's the difference? Usually the difference is other features such as the number and type of focus points used ... or how quickly the camera can rapidly burst shots ... or the size of the camera's internal memory buffer. If you're doing a lot of action photography, there are features a camera might have that optimizes it toward action photography. But if you're shooting a landscape on a tripod using a remote shutter release and you have all the time in the world to get that shot... having loads of auto-focus points and high-speed burst isn't really going to help you. On the other hand if you are a sports/action photographer ... the lack of those features might mean you get fewer 'keepers' as the camera struggles to have the focus system and shutter keep up with the action.
The above is my priority list. A different photographer might give you a different order. I don't get to hung up on brand names or equipment. Loads of companies make fantastic cameras. It is possible to select a camera that may not be ideal for a particular type of photography ... and it is possible to select a lens that isn't ideal.
Ultimately the ability to have one thing in sharp focus and another thing far out of focus (to make that thing blurry) is based on an idea called the Depth of Field (you'll often see this abbreviated as DoF).
If my subject is 10 feet away and I point the lens at the subject, we'll need to focus the lens on the subject. In reality we are adjusting the focus for a 10 foot distance. If something is not precisely 10' away... suppose something is 9'11" or 10'1" -- will being fractionally nearer or farther make a difference? Probably not when you consider the ratios... 10' = 10 x 12 inches or 120 inches away. So a 1" difference works out to a difference of just 0.8% (not even 1%). But what if a lens was focused to 10' and something else was 100' away... now the difference is more significant.
DoF is the idea that there's a range of distances at which you will judge the subject to be ... more or less ... acceptably focused (I didn't say perfectly focused). How you judge this will also be affected by how closely you inspect the image.
There are a few factors that ... when added together ... affect the overall depth-of-field.
This include things such as:
Focal length of the lens where short focal length lenses tend to produce much broader depth of field and very long focal length lenses tend to produce much narrower depth of field.
Focal ratio of the lens. The focal ratio considers the size of the physical opening in the lens through which light may pass. But instead of being a simple diameter ... it is expressed as a ratio of the lens' overall focal length divided by that physical diameter. If a lens with a 100mm focal length has an aperture opening 50mm across than that lens has a focal ratio of 2 ... since 100 ÷ 50 = 2. That would be expressed as f/2 (the 'f/' is short-hand for focal ratio). If we adjust the opening so that it has a diameter of 25mm then the focal ratio becomes f/4 because 100 ÷ 25 = 4. One take-away is notice that the lower-focal ratio example (f/2) had the larger physical diameter opening in the lens (50mm opening on a 100mm lens). If the lens had a very tiny opening (a pin-hole) then all the light has to pass through just that one very tiny point. Such a lens would have and extremely large depth-of-field. If you use a lens with a very large opening, the photons have a choice of many different paths through the lens. This reduces the overall depth-of-field.
Subject distance is another factor. If a subject is very far away a lens needs to be focused to near the infinity point. Of course everything in the background is even farther ... so also focused to near the infinity point. Since there isn't much difference ... the background seems to be more or less just as well focused as the subject. Foreground subjects might appear blurry. But if the subject were close to you... and the background was quite far away, the background will probably appear to be more blurred. As a photographer I rarely place a subject against a wall for a portrait shoot... unless that wall is extremely interesting and will add to the overall value of the shot. Better to pull them away from the wall so that the wall can fall begins to get farther outside the depth-of-field.
All of these factors have to be combined... each one simply influences the overall depth of field. None of them completely rule the depth of field.
A camera with a long focal length lens using a very low focal ratio and a subject placed somewhat close to the camera with a background placed much farther away ... will result in strong background blur. Doing the opposite ... short-focal length lens, high focal ratio, and a more distant subject ... will produce very large DoF and you won't notice much blur in the background.
Other Considerations for Blur
There is a term used in photography called 'bokeh'. This term refers to the quality of the blur... not the strength of the blur.
Once upon a time, Canon produce a particular 50mm f/1.8 lens that had merely 5 aperture blades inside the lens and these were not well-rounded blades. This means the opening in the lens was a pentagon instead of a circle. As points of light blur, they blur with the geometry of the aperture opening. This means that as pentagons overlap pentagons it created a rather strange effect ... the quality of the blur was not very smooth. It was somewhat jagged ... jittery ... almost nervous looking. It did not evoke emotions of peace and beauty that will add to the effect most photographers were going for.
Canon replaced that lens with a lens that has identical optics (not similar... identical!) But that newer lens has 7 aperture blades instead of 5 and the blades on the newer lens do a bit better w.r.t. to being more well-rounded. This produces a more circular opening and the results in images produce a much better quality in the background blur (the blur appears more smooth).
The lenses are optically identical and they have identical focal ratios. But one lens produce a poor quality blur. The other produces a higher quality blur.
This is what the term 'bokeh' is meant to convey... the quality (over quantity).
This means that even if you combine the conditions such as focal length, focal ratio, and subject distance (relative to background distance) to try to maximize the background blur... it is possible to have a blur ... but a blur that you may not enjoy.
- An entry-level DSLR body is fine for portrait work. Portraits usually do not require advanced camera features.
- You'll need to pair it with an appropriate lens to achieve the results you want (blurred background portraits). A DSLR is often packaged with a 'kit' lens and this is usually a standard zoom offering a bit of wide angle and a bit of narrow angle but not a particularly low focal ratio (manufacturers try to pair the camera body with an affordable lens to put them with financial reach of more consumers). Using a 50mm lens would be better ... using an 85mm lens would be even better still.
- Your skill will be very important ... the equipment alone wont be enough. If you use the 'automatic' setting mode it will tend to go for 'safe' exposures ... not artistic exposures. Skill is needed to learn which settings produce the results you want.
- Lighting will also be an important factor. Good lighting isn't just natural lighting... it's lighting that does what you want. Photographers with deeper pockets invest in equipment (although a lot of lighting gear can be surprisingly cheap when you compare lighting costs to lens costs and other equipment costs) ... but they invest in this gear specifically because it lets them control the lighting.