I roleplay in play by post games, both syncronous (play by chat, everybody has to be there for the game to progress) and asyncronous (play by forum, play by e-mail).
The first thing you need to realize is that if you try to play with the same rules of your weekly, we-meet-at-Bob’s-on-Fridays game, even a synchronous game will be really slow. Espsecially if your players like to be detailed (in my games we write abbout one post every 5-10 minutes depending on lenght, and with longer posts it is often a matter of telling people “before he does that, I say this, and then I say this after he finished”, which will make you crazy if you ever try to find out the timeline of when everybody did what they did.) So slow that you won’t be able to play many backdrop events at all, and the backdrop will lag behind.
Since a play by post game is that slow, my suggestion is to have a chat (which is better than a forum in my opinion because it’s easier to see if someone is writing, avoiding crossposts) and just brainstorm what should be happening, instead of outright roleplaying the whole thing.
Ignore resolution mechanics, go with the flow and maybe use a chat like the one provided by Discord where you can comment with small icons under each sentence, like the new Facebook emotes that got introduced along the usual “Like”.
Paint the world at large, deciding together what you, as a group, think is going to happen in the game world as the result of recent events (both decided here and as the result of your characters’ actions during the usual game sessions).