He wants to become famous at all costs. He doesn't care how or what for. He just wants to become famous. He doesn't care if he becomes a singer, dancer or actor as long as it is one of the 3. He's been taking singing, dancing and acting classes since he was little.
In the above Are the following correct? He doesn't care how or what for.
"At all costs" doesn't really fit this situation. No matter what else, he wants to become famous. It doesn't matter how, he wants to become famous. He wants to become famous, no matter how. "He doesn't care if....." is good.