You level up in Demigod and you start every game with level 1 again (I doubt you will start with level 0
)
The only thing which remains from game to game are solely of aesthetical nature : your Achievements (like being Top 100 in ladder or having won 100 ranked games or whatever the achievements will be) will be visible as non-gameplay-affecting monuments (maybe they will also affect the look of your Demigod and/or other things but I am not sure about that)
(check GPG forums if you don't believe me)