To increase the amount of background blur, you can do a number of different things:
- Decrease the distance from the camera to the subject (shoot close; this is why macro shots have very thin DoF).
- Increase the distance between the subject and the background.
- Increase the focal length of the lens you're using.
- Use a wider aperture setting.
And, of course, some of these are going to be opposed to each other (longer lenses may make you move back or force you to use a smaller aperture setting). And may still only get you so far. But aperture isn't quite as all-powerful in this situation as you might assume.
So, getting a new lens alone may only get you so far, particularly if you're only looking at max. aperture. Just my guess but your example image may have been shot at a smaller aperture than you think with a much longer lens than you think (say, a 70-200 f/2.8 vs. say, an 85/1.8 wide open).
Using a larger sensor may help you get more background blur more easily. The larger sensor itself doesn't decrease DoF simply by usage, but because to get equivalent framing vs. a crop body, you'll probably be closer to your subject or using a longer lens (or both).
I've gotten a good amount of background blur on a crop body using f/5.6 with a 400mm lens, and only a minimal amount of blurring with a 50mm f/1.2 lens wide open on a full-frame body, with a more distant subject. So it does depend on a balance of all four factors.
Another technique you can consider (if your subject isn't moving) is to do a bokeh panorama, also known as "The Brenizer Method". This is where you use a long fast lens on a camera, but shoot with the subject closer in portions, and then stitch it together in a panorama, thus "faking" a larger sensor. A lot of full-frame shooters do this to fake "the medium format" look of thin DoF at longer subject distances.