Cheapest is to use a smartphone or tablet with a good camera that you already have. Use it with frame grabber Stop Motion Studio. Most have pretty good HD video resolution, some 4k, and if you get one of those phone holders that mount onto a tripod, results can look ok. Stop Mo Studio can lock off the phone cam's focus, exposure, and white balance so they don't keep changing while you animate, which is essential. Not sure if there is depth of field on any of the latest phones, my iPhone SE or iPad don't do it.
Best, and same as feature films use, is a Canon EOS DSLR, ideally with 2nd hand manual lenses from Nikon, Olympus, or other maker, with lens mount adapters. (The Nikon 55mm macro is good for close-ups.) That definitely lets you set the aperture and exposure time to get depth of field control. (Canon lenses, and the kit lenses that come with DSLRs generally, do not have aperture control on the lens, so are not fully manual, and if the camera is controlling some functions it can cause flicker from trying to "improve" the exposure all the time. Using another brand of lens isolates it, so settings stay where you put them.)
There should be plenty of DSLR cameras around for sale that are not the latest, but do have Live View, by now. I like my old 40d, and slightly less old 7d, but there are also more entry level models like the 600d that give great results.
You would also need a frame grabber on the computer, like Dragonframe, which would also cost money US$295 (UK 222 pounds) which is already too much. Look at the Dragonframe website and see what cameras are supported - it is really dependant on the camera maker making it so it can be controlled remotely via USB cable on a computer, so if the camera is in the DF list it will likely work with other framegrabbers. https://www.dragonframe.com/camera-support/
Others may know of less expensive options - unfortunately I did my learning back in the old film days, so I only have experience with pro options in the digital era.
If you had to buy everything - DSLR camera, AC adapter, lens, capture software- it would go way over budget. But if you happened to have a tripod, a suitable laptop, and some old lenses for 35mm still cameras in a closet somewhere, a cheap adapter or two would let you use them. There are lower cost frame grabbers, but I am not up to date with those. So the DSLR option is only affordable if all you need to buy is the camera.