As above it depends what you want to check.
Most diagnostics are visual (is there a big sooty mark, things pulled off, cracked board...), parts swapping/elimination (make minimum viable machine, try from there) or whatever the OS tells you (while not super reliable then most things do have onboard checks and will tell you what is wrong).
For laptops I have occasionally poked the power supplies with a multimeter and found them broken.
Several times for older computers people still want running (runs a particular bit of software, piece of hardware...) then I have replaced capacitors on motherboards and graphics cards. 99% of the time that is a visual thing but if doing that properly then you would really want a equivalent series resistance/ESR meter, which I don't think I have ever seen on a multimeter (though cheap ESR testing devices are available these days, sometimes even included in component testers) and I am not really expecting to any time too soon.
Could have done the same for power supplies as well but 99% of the ones that fail are cheap and nasty anyway, and I have a pile of the things from computers I have scrapped.
Have soldered up some cables to do some more custom supplies for fans, lights, things where one port was broken, cooling devices and such like. Can help there to find what is delivering what and what you might want to do with it.
Exceptions tend to be for Apple products. Here they don't sell parts to you if they can help it, and as they are so ridiculously expensive (both new and second hand) it becomes viable to do component level replacement/board repair. In such scenarios a multimeter is an essential piece of kit.
I would think long and hard before getting into apple repair as it can be quite a setup cost, and parts availability is always a nightmare. Most would probably point you at
https://www.youtube.com/user/rossmanngroup/videos if you were interesting in such a path.
I have also resoldered charge ports on laptops, or indeed adapted them for new ports if replacements are not viable/available, though that is usually a visual thing (I have confirmed no power getting through though with a multimeter) as well but if you want to test things/make sure you are putting the power in the right place then helps to have a basic one.
Again though 99% of computer repair is either taking parts out until it boots, checking things are plugged in properly, swapping parts around into a working computer to see, swapping out parts with shinier ones.
If you want to get a cheap one that tells you what voltage is coming out and whether there is a connection between two points then great. I would not bother with a fancy one for basic computer repair and instead make sure I have a nice set of screwdrivers as those will be far nicer -- you will be inserting and removing screws hundreds of times and ones that cause cramps in your hand are not fun.
