There are several RP Fleets around but before you join any of them I'd encourage you to check out the Alpha Centauri Institute.
We are a mature (drama-free) RP fleet with a strong core and a growing membership. We do in-game RP centered around character stories, PvE, and our own "episodes". We will be taking full advantage of the Foundry once it is released.
If you are interested, please visit our website at http://aci.guildportal.com for more information, check out our recruiting post here in the Starbase One forums, or drop by Starbase 39 to see members of our fleet in action most evenings ET.