What is the quickest way to level up?
From the wikia wiki:
Quick Play awards more XP than vs AI, here are the values for various actions:
- First Match Win of the Day: 1500XP
- Win: 500XP
- Match Finish: 250XP
- Time: 200XP per minute (the fextralife wiki reports 4.01 per second instead, which works out to 240.6XP per minute)
- Consecutive Match: 200XP
- Gold Medal: 150XP (only highest medal achieved applies)
- Silver Medal: 100XP
- Bronze Medal: 50XP
- Group Bonus: +20% EXP
- Leaver Penalty: -75% EXP
Given this list, it looks like your best bet to is play in a group for consecutive winning matches, while getting a gold medal in each match. Medals can be awarded for Eliminations (Overwatch's word for kill participation), Objective Kills, Objective Time, Damage Done, and Healing Done (H/T SaintWacko at this question). Only one medal bonus will be applied per match, for the highest medal earned.
There is also an experience bonus for backfilling when you get slotted into a match-in-progress in place of someone who left, but that will be more difficult to deliberately cultivate.