Thanks for playing! :)
Essentially you get XP when you kill an enemy; around wave 6-7 your character should probably be about level 6-7 (although the number of enemies spawned during a batch of a wave and the XP reward for a killing an enemy are both randomized slightly). You get one upgrade point when leveling up. The "Upgrades" button only shows up at the end of a wave if you have upgrade points to spend.