Dear Vinyl-Dye Anon
(Link related.) What you want is Jacquard i-Dye Poly (not regular i-dye!) for the vinyl parts and Krylon Fusion spraypaint for the joints (because plastic doesn't take dye for shit.) If you can't find i-Dye Poly, you can try RIT Dyemore. Right now everything is hard to track down, so you might have to hunt through multiple storefronts to find the colors you want. Jacquard has a listing of webstores who carry their products -- eventually you'll find them!
Get a steel or ceramic pot (aluminum can react with the dyes) you don't intend to ever cook in again, bring the water to boiling, and (wearing gloves) add in the doll parts. The longer you leave them in there, the darker they'll get -- I recommend dyeing darker than you want the doll to be, as vinyl will lighten over the next few weeks to its final color. I recommend getting some extra vinyl to sacrifice for testing before you dunk your actual doll, because colors can be very fiddly -- browns commonly have a wholly unflattering green undertone, and you'll want to mix different amounts of warm colors to get the actual shade you want. (To that end I also recommend getting a bunch of tiny measuring spoons and storage containers so you can actually make yourself a dye recipe for the colors you want, in case you revisit it in the future. This also allows you to store the extra dye for later, as you rarely need the entire packet of powder for small-scale jobs like this.)
Once you've got the vinyl in the color you want, go outside with a mask on and spray the joints into whatever color Krylon Fusion is closest. you will probably have to do touch-ups now and then, c'est la vie -- you can try dyeing the frame, but your mileage on results will vary tremendously between "nothing happened" and "saturated magenta elbows", so be careful! I dream of the day companies just... sell vinyl in more colors... easily... without having to jump through hoops... or physically go to Japan... or beg, barter, and grovel as with a certain well-known vinyl dollmaker... until then, you can absolutely dye a doll, and more people should! Let's get fantasy colors in on it too. Have fun!