How can I tell how many points I’ve earned in a competition?

To see your and teammates' individual scores, tap on the the competition calendar icon, then tap on the green button under your team's score. This is the same green button that shows how many players have played in the competition over total number of team members.

If you are referring to how many league points that you have earned during a competition, there is no place that shows that. You'll have to look at your league points prior to and after competition and do the math.