The offseason is in the summer. If there is the Euro or World cup, then the leagues finish a couple of weeks earlier and start a couple of weeks later.
Usually the teams that are in multiple competitions will add more depth which is the challenge. International Milan a few years back won the treble (serie A, Champions and Coppa Italia) which is why they'll go in the history books as one of the best teams ever. Generally you have to finish in the top 3-4 of your league (depending on your country) to qualify for next year so a team like Man Utd would be considered a disaster if they didn't so it is all about always finishing in the top of your league as well as taking that little extra risk for the champions league.