Well, the first question you need to solve is the magnification you need. I've seen comments talking about reverse lens setups, but those are really for extreme amounts of magnification, and extreme amounts of magnification come with comparatively shallow depth of field.
Taking a look at the video you referenced, the field of view appears to be a bit more than 6cm across (the connector grid is likely 2.54mm, so the whole connector is about 5cm across). Let's say 7.2cm because that gives us something easy to work with. A full-frame sensor is 3.6cm across, so you need a magnification of 0.5×.
Now let's assume that you use a superzoom like the Panasonic DMC-FZ330 (not a recommendation as such, just a camera I am familiar with). The advantage of a small-sensor camera is that it tends to squeeze the minimum aperture size out of the specs: here the minimum of F8 corresponds to F45 on a full-frame.
Macro mode is not a good fit because of the distance of the lens blocking sensible attempts at getting light where you need it, so assume rather tele mode to get the magnification. We have an effective(!) focal length of 600mm which means that for a magnification of 0.5× (when assuming an "effective" sensor width of 36mm corresponding to the effective focal length) a distance of 1200mm is sufficient.
Unfortunately, at 600mm effective focal length, the minimum distance for the FZ330 is 2m. So you need to engage closeup lenses. For good image quality you want achromatic closeup lenses. Of those there are lots around. For our task here, even a +1 dioptre will do the trick. It transforms the camera's idea of a distance of ∞ into an actual distance of 1m, and the camera's idea of 2m into an actual distance of 66cm. A +2 dioptre lens would map the camera's distance idea of 2m–∞ into 40cm–50cm, a +3 dioptre into 28cm–33cm. There really is no point in going even stronger since both working and focusing range will become increasingly awkward to manage for a camera mounted on a tripod.
A +2 dioptre (like the somewhat overpriced Panasonic closeup lens) with its minimum focusing distance of 400mm at 600mm effective focal length will allow us to go to 1.5× magnification in reference to a full-frame sensor, meaning that the field of view will correspond to 24mm across, or a bit more than 9 solder pads of distance given a 2.54mm raster.
Now in general you will want to use the minimum or base ISO setting on your camera, and you will likely want to close down aperture a lot. At video resolutions, diffraction is usually not much of a problem, so closing down aperture is usually an obvious choice for getting good depth of field. The combination of closed down aperture and low ISO means that you will want light. A lot of it. Let's assume that you want to make web videos at 30fps, then the usually desired exposure time for "cineatic" movements is 1/60s.
For bright sunlit scenes, there is the "Sunny 16" rule. At ISO100 and 1/60s, it would call for an aperture of F21, about 2.7 stops narrower than the F8 we are aiming for. That sounds like we have a lot of reserves, but the usual in-door illumination is actually quite massively less than bright sunlight, so you need to plan for considerable amounts of light as well if you don't want large motion blur trails and/or lots of mottled image noise by using longer exposures and/or higher ISO.
The usual advantage of a large sensor camera, namely that it will support quite higher ISO values and thus lower light situations pretty well, does not actually pay off once you need to stop down so much that you get similar depth of field as with a small sensor camera. So there is not much of a point in springing for a larger sensor camera. Instead, put aside enough money for getting your comparatively small field of view illuminated like an almost bright day outside.
The math may end up different for different camera models, and possibly you'll even find one which has a useful magnification (due to a lower minimum focusing distance and/or longer effective focal length) without employing closeup lenses.
More likely than not, the built-in microphones will point somewhere entirely useless for online commentary. So either you create your sound track offline and/or or you factor in some external microphone(s) (and make sure that you picked a camera with external microphone input) or a strategically placed sound recorder. If you don't record your final sound on the camera but on other devices, the built-in microphone(s) will come in handy for synchronising your sound track and the quality is not much of an issue.
For external sound setups, you'll need some video editing software, too, and that also makes sense since single-take videos tend to strain the attention span of the listeners.
Note that all this is aimed at producing stuff of high visual quality that does not burn through most of its encoding bitrate by encoding image noise. Indeed, a high quality source with confined amount of movement and low image noise will actually render quite well even at lower bitrates.
