The Nintendo Game Boy Color does not have a built-in charger or rechargeable battery. It runs on two AA batteries (alkaline or rechargeable NiMH).
Use Eneloop Pro (2550 mAh) or similar low-self-discharge NiMH AAs + a quality charger (La Crosse, Panasonic, Nitecore). Carry spares. Avoid cheap "GBC battery packs" with proprietary cells—they age poorly.