You can join a Private League at any time. You can join multiple Private Leagues during a season.
As long as you are not currently in a PL, you can join a PL.
A private league will finish at different times depending on how many teams there are in the private league and what the settings are for the private league. But as soon as it finishes you can join another PL.
Your supporter must last longer than the length of the PL to be able to join it though.