Does the English Premier League and the UEFA Champions League run at the same time?

[Soccer-Novice]

They way I understand it is a team from EPL (or any top European League) can qualify for the UEFA Champions League by doing well in their own league. If they do qualify, do then compete in both leagues at the same time (EPL and UEFA). Do they choose which league to compete in? It seems like a brutal schedule to compete in both leagues at the same time. I'm just starting to get into to the EPL and I find it very confusing the way teams can be delegated and so on, much less qualify for the UEFA. I think its a great idea, one they should use among sports teams here in the U.S. It really seems to find out the true Wolrd Champion.