*Update February 2021: Joining at a specific time doesn't matter any more. Duolingo seems to be placing people in the same league based on their point totals from the previous week.
Update July 2021: Duolingo has changed the amount of xp awarded for each practice of a gold circle from 20+ xp down to 10+xp. They have introduced the legendary level. This legendary level is only available after you have reached the 5th level for a particular circle.
This legendary level is probably not available to everyone as they do A/B split test it seems.
Update September 2021: Depending on the A/B split tests, Duolingo offers challenges to complete during the week to earn 40xp. There are lightning rounds and timed practices you can do during the week. They both offer up to 40xp. You can win leagues with these 40xp challenges compared to 20+xp for doing hard practice with crowns and legendary levels. Crowns have become less effective for league play.
Use duome.eu/username to check on your competition.
example
www.duome.eu/pspoon2727
Scroll down until you see the section called "Daily Progress"
Next to the "Daily Progress" Section you will see a grey button called "raw"
Click on the teeny tiny grey button titled "raw"