By request! As it's superdeath early for me, this guide will be quick and subject to serious review/rewrite later.
Forum RP 101
There are a few key aspects to keep in mind about forum RP that differs greatly from in game RP.
FOLLOW THE LEADER
Most forum RP stems from a particular plot event or incident that's been building up in game. It usually revolves around character issues or an environmental problem, and as such it's either actively DM'd or 'almost' DM'd by someone in the thread who has some idea how they want it to end. It's fine to throw In Character loops or concerns, but overall don't go out of your way to change the final big outcome of an event. I.E. If someone is sick, and you're playing the doctor, don't throw out an 'Oh NO, you're going to die' when all they needed were glasses.
Because forum RP generally has a desired outcome, it's imperative that forum participants communicate. If you're the 'DM', make sure the participants generally know where you're going. If you're a participant, make sure you're clear on what the limits are so you aren't the doctor telling a man with a bugbite that he has cancer. Communication ALSO lets you bring in some fantastic insights and post flow that you might otherwise not have. This is hard to do without knowing your forum partners well, since it can look like meta, so make sure everyone is comfortable before trying this 'advanced' technique. (Forum RP 201?)
I SEE (AND HEAR) YOU
In real life, a conversation has all sorts of cues- intonation, volume, expression, body posture- and not all of them are intentional. When forging a forum post, consider the five senses and give your partners a rounded view of what your character might be thinking and feeling based on those cues-- including the ones your character might not even be aware of!
WHO'S A GOOD BOY?
Forum takes longer to write and should have more detail (see style), so make sure you're throwing some good bones to your partners. Don't make one sided responses or only answer the questions asked of you (unless that's your character)-- you'd be surprised on what conversational gems can stem from 'natural' conversation flow. If your character isn't loquacious, supplement that lack of conversational tangents with physical ones. Maybe they fiddle with their coffee mug. Maybe they shift their weight when they lie. Give other characters *something* to pick up on, whether it's an off handed comment surrounding the context ('Yeah, I had a girlfriend like that once. I'd say give it time.') or a physical tick/habit.
Be prepaaared~ for a time commitment. Forum RP takes extensively longer to play out than in game RP, for obvious (it takes a while for someone to write up a post and reply) and not so obvious (it can take time to come up with enough content for a good reply) reasons.
As forum RP takes so much longer than something in game, it's common for participants to agree on a time and duration for the event to have occurred on the 'in game' timeline and be somewhat vague about the details of a scene until the forum RP is actually concluded. For example, if someone is in sickbay and everyone generally agrees that the character is *not* going to die, the character in question might either keep their IC head down in game (depending on the amount of suspense they wish to generate), or just RP as being 'in recovery' from the main forum event. The exact terms of a Time Warp are negotiated between the forum participants, so communicate!
Forum posts tend to be longer, more descriptive, and even sometimes offer insights into a character's thought processes-- especially if those processes are reflected in the character's expression, words, or posture. While 'internal thinking' is usually a no-no for in game roleplay, it can serve in forum RP to set the tone and offer 'instinct' insights to other characters involved. Don't go overboard with thoughts players can't see, but sprinkling one or two in next to external cues can enrich the setting. (I.E. 'That's because Razor's a jerk', Chassy thought instinctively, her nose wrinkling with displeasure at the thought. But aloud, she only said: "He can be, ah, difficult.")
Since forum posts give you the benefit of time, take it. Reread your post before you hit the send button, maybe even read it aloud to test the pacing. Run it through spell check. Ensure that it's comprehensive; your RP partners will thank you for it.
FOUR'S A CROWD
The best number for a forum RP is around three-- too many people, and you can get stuck waiting forever on one person's post and the RP is liable to be forgotten altogether. (Don't feel bad, this happens from beginner to expert! It's just a fact of forum RP.) Or, alternatively, someone feels left out if you continue without them because they took two days or so to get a reply in.
1 + 1 = AWKWAAARD
Too few people, and you might be stuck grasping for content if something unexpected happens, or you get caught in a conversational rut, ICly or OOCly. Three is a good number to give everyone a little break from the 'must respond now' responsibility without forcing the creative juices in a tight spot.
KNOW WHEN TO QUIT
Nothing should last forever, even the good things, so make sure you have an end-goal in mind that ensures a resolution before the forum RP is feeling drawn out or too dry. It's better to err on the side of 'too short' than 'too long', as a forum RP that takes ages to finish can leave a bad aftertaste that taints even the best of starts. Don't be afraid of communicating with your RP partners about when you think you should start wrapping things up,
Feel free to comment/ask questions/request clarification/rage and offer alternative opinions.