does it affect the character if i max it out in level first and then use the keys upgrading the nodes to get the highest power level? or is it better to level up the skill tree while leveling up fighter level.
Nope. Only thing that makes a difference is if you're upgrading a gold vs a bronze fighter, the gold is gonna take more coin and skillpoints.
Honestly I'd go for upgrading as you level since you're gonna need your characters to be beefed in order to get a high Prize Fight streak, which you need for just about anything as well as leveling.
It kinda just comes full circle if you upgrade while leveling rather than waiting, you'd be wasting a lot of time if not.
Character level and character skill tree are totally separate: Feel free to totally max out a skill tree with all keys and nodes at level 1 if you want! I tend to just unlock as much of the skill trees as I can so that I don't have to worry about it later, but just upgrading as you're leveling up is totally fine.