The Xbox 360 wireless controller uses two AA batteries — no built-in rechargeable battery. Options:
Buy quality NiMH batteries (Eneloop, Amazon Basics) and a separate charger. Cheap, flexible.
Similar to Play & Charge Kit; often higher capacity. Charge via USB.
Use a USB data cable (not charge-only) to play wired while charging any rechargeable pack.
Tip: Avoid alkaline disposables — expensive and wasteful. Rechargeable AAs or a Play & Charge Kit are best.