Gotenks is available through the capsule shop in Dragon Ball Z Budokai 3. Obtaining Gotenks requires the black membership card, 30,000 zeni, and entering the capsule shop with Goten or Kid Trunks.
Because the contents of the capsule shop are random, unlocking Gotenks can take quite a while. There is no easy method or short cut beyond visiting the capsule shop often using either Goten or Kid Trunks.
The black membership card, required to acquire Gotenks, is available in Dragon Universe. If Goku is level 30 or above, he can locate the card on the island Northwest of the main continent.