Playing Skylanders Online on Xbox 360: Not Possible (Local Co-op Only)
Skylanders games on Xbox 360 do not support online multiplayer. The portal-based toy-to-life system was designed for local play only.
What You Can Do
Local Co-op (2 Players): All main games (Spyro's Adventure, Giants, Swap Force, Trap Team, SuperChargers, Imaginators) support drop-in/drop-out couch co-op. Requires two controllers + two portals (or one portal, swap figures).
Figure Trading: Swap physical toys with friends to unlock their characters in your game.
Leaderboards: Some games have online leaderboards for time trials/score attack (async, not live).
Online Play in Later Games (Non-360)
SuperChargers (Wii U, PS4, Xbox One, iOS): Online racing co-op (2 players).
Imaginators (Switch, PS4, Xbox One): Limited online co-op in Sensei realms.
Ring of Heroes (Mobile): Turn-based RPG with PvP — different genre.
On Xbox 360: Couch co-op only. No Xbox Live multiplayer.