i only know how to do independent rp (i.e., not in a game) on tumblr since i've never joined a game there, so if anyone wants to chime in with game-specific tumblr rp advice that'd be cool. but for independent rp:
make an account for your character. some people make multi-character accounts, but i find it easier to start with one and then add more later as you get the hang of it. it's best to make a new account for this because a sideblog on your main account can't send asks or follow people, which are key parts of tumblr rp.
set up your account with a bio of your character, essential information that you want people to know about them. a lot of people also make ooc guidelines, which are just like basic rules for rping with you - guidelines with what you are and aren't comfortable rping, basic stuff like "please don't godmode or hurt my character without asking me," all of that.
then start posting open starters. this is a general piece of dialogue or narrative piece of your character doing something, just like how you'd start an open threadstarter in a journal game. you want to use (and track!) the tags #indie rp, #independent rp, #open rp, etc. (i'm pretty sure tumblr only counts the first five tags you use, so i make the first five rp-related tags and the rest whatever personal navigational tags i use.)
there are different kinds of "threads" on tumblr. there's: - gifchat or dashchat: characters just chatting with each other with minimal narrative cues, usually accompanied by gifs of their pb (called face claims/fcs on tumblr) illustrating their expression etc. (if you need gifs of your pb, search tumblr for "[celebrity name] gif" or "[celebrity name] gifhunt"). a gifchat is like normal commenting between characters in a journal game or as if they were chatting over aim or something, but usually like they're standing face-to-face. people sometimes use brackets or asterisks for short physical actions. - para: short prose posts, also called storybook. usually one or two paragraphs. may or may not be accompanied by gifs of the pb. - multi-para: same as above, but usually much longer.
track the tags you use for rp like #indie rp #open rp etc. also, track your own url. that'll help you see posts people make for you that they tag with your url. (it's best to pick a url with no hyphens in it, because tumblr does weird stuff with hyphens and tags.)
then browse the tags and reply to other people's starters. if you see characters that look interesting, reply to them and/or follow them! if people follow you, it's a good idea to post greetings - having your character say something like "hey there, what's up" and tag the other person with their character's url. (their blog usually says what tag they track.) don't get discouraged when people don't reply to your greetings. it happens a lot. i stopped doing them because it frustrated me since i put effort into them, but it's the best way to connect with new followers if you're too shy to send asks like i am.
speaking of that: send people asks! most people love to get random people in their ask saying hi and starting in-character conversations, or ooc asks discussing the potential to make plot. sending people asks is the quickest way to make it clear to them that you really want to rp with them and it also shows your url to all their followers + indicates your willingness to send asks to them. i find a lot of people to follow that way.
i recommend stalking the tags #indie rp and #open rp for a while and stalking interesting-looking blogs you find there. observing other people is a good way to get the hang of what's going on there.
also, when you get in a thread with someone, make sure you "cut" your posts by deleting all but the last two or three exchanges. when posts have 9000 reblogs and start to get squished on the dashboard it is extremely annoying and everyone hates it.
you can also search tumblr for "rph" accounts or "roleplayhelp" or "helper" accounts, they're there to help rpers with any questions they have, with graphics and layouts and finding games or people to rp with.