So you've applied for a character and had it approved, and you're all setup with your bright, shiny new SSet character. What comes next? Well make sure you play the character. Being active is a great way to show the Immortals that their faith in you was well placed. Also, remember that section in the application about what you want to do with the character? Well make sure that you actually start trying to do that. That is part of the reason why the Immortals chose to approve you for the character, so be sure you haven't lied to them.
Also, you should note that application characters pretty much remain the property of the Immortals. If they feel that you aren't really playing the character often enough, or well enough they may take it from you and offer it to someone else. So if you were supposed to be teaching axe/dagger/ashandarei forms, then head out there, find some worthy pupils and start teaching. If you were supposed to be an Aes Sedai, then be sure to go play a Sedai, and play it well. Especially with regards to the Ajah you've chosen. The Immortals will likely offer some advice or warn you if they aren't particularly satisfied with your performance. There are certain things they expect out of application characters.
On the flip side of things, perhaps your application was denied. What does this really mean?
Well there are a couple possibilities, the first (and most common) of which is that it simply wasn't well thought out enough. Rethink your background, and your concept for the character. Tighten up the language, spelling, grammer, etc. and consider applying again! You might even want to sit down with the Immortal who reviewed your application and ask them to point out places they think it could have been improved, why it was denied, etc. More often than not they will be willing to do that. Another thing to consider is that the Immortals might not have seen enough of your ability to roleplay. That's one of the key factors that will go into an application's consideration. So go out and roleplay, do it a lot and with a lot of different people. Submit some roleplays, ask the Immortals to sit in and watch, perhaps try to get in on some events that they are running.