You can use the character on and offline. I think they level up separately though.
Oh yes, you level up your skills. You gain xp for games. There are three different skill trees, offense, defense, and athleticism. Depending on your performance and your class, excuses me, position, you gain xp differently. For example, my power forward is gaining xp faster in athleticism, but a sniper would gain offensive xp faster. Performance is a big factor too, when I get straight A's in my performance I get a nice bit of xp, but it goes down if my performance does.
It's also a slow progression, each skill takes about 200 xp to raise, and you gain about 10-20 xp a game. There are a ton of skills too. I need to up my accuracy, my one-timers are waaaaay off.