How do I unlock seasonal units?

Seasonal units are unlocked in your V-Menu, under Seasonal Challenge. Use the top right to swap to older seasons, then complete the challenges of the unit you want to unlock it.

How long it really takes

There is no honest number of weeks, it depends how hard you push. What most people do is play four or five games per challenge page and pay for the rest. That is by far the least annoying way to do it.

Only some of the quests are worth actually focusing on: the ones that want a number of kills from your units, or a PvE win. The rest turn over on their own while you play.

See also: the V-Menu and every menu explained and the season system.