I want a character to have an ability at level 1. When he levels up, the skill gets upgraded to level 2. It will have different properties than the first one and the first one will be obsolete. How do I make the character get rid the first ability?

Hmmmm.....since you didn't say I'm going to assume RM3....

There should be an event command in the 'party control' section of event mapping that says 'lose ability'. Just make an event that erases it. Granted there won't be any precise way to make it lose the ability exactly when you level up (not one I know of), but you could probably figure when they'll hit a certain level-and put the event there.
That should do it...unless you aren't on 3.

Oops! Yeah, I'm on RM3.

I upgrade skills after Boss fights. They are event battles and the opportunity is easy to use.

I also just use more powerful versions of the same skill with the same name so the player never notices except that his skills are getting stronger. I remove the old ones and add the new ones just as described above.

Peace.