Xbox – play Destiny on the Xbox one against the brothers who both have Xbox 360

All you need is the game,internet and Xbox live gold membership.

To start it, click "Xbox live" in the main menu.
- Click custom games or quick match to join someone else's game. Pick the match you want to join and then it will ask you to choose your character. Everyones matches to choose from will be labeled by what level character they are and what quest they are working on.
- Click Create game to start a game with your character.

All you need to have is an Xbox memory card to do this:

You need to transfer an Xbox Live account to that Xbox in order to play. The reason why you need an Xbox Live account is because the whole serial code was not needed for Xbox it only required that you had Xbox live or an old account.

You have to have a gamertag to play Xbox PSO, even offline. Xbox Live for the original Xbox was shut down years ago so you can't create a gamertag anymore. If you have a means of copying a friends gamertag to your Xbox that can work. If not, you should probably consider playing on a different platform.

Sources: Sega Forums | Phantasy Star Online Forums