In World of CHEL, it will take you to level 32 to unlock all traits, specializations, etc. that affect your gameplay. (Aesthetic unlockables will not necessarily be unlocked by this point.) A fast way to unlock things is to play against the AI in the offline mode. The AI is set to the easiest difficulty setting, so it is very easy to beat them, allowing you to win games quickly and level up fast.
The ability to use jersey logos for your EASHL club is a random unlock. You might get it after a few games, or you might need to keep trying for a while. While some unlocks come at set levels, this one appears to be completely random.
As you play through Threes Circuit, you will have the chance to unlock new characters for use in Threes Circuit, either offline or online. While you can win a match on any difficulty level to continue through the game, to unlock a character, you’ll need to play on a specific difficulty setting and earn three stars.