The main way to participate in the Altador Cup is to sign up for a team during the Sign-up Period, and then play games during the Tournament period! Playing games will earn you Prize Points, Neopoints, and will contribute to your team's overall score to determine where they place in the Altador Cup.
If you miss the sign-up period, however, don't fret! You'll still be able to play the games and earn Prize Points, your scores just won't contribute to any specific teams.