Answer: Betty won, then Dave, Adam, Charlie and Eileen.
By placing the person who finished higher to the left we have:
1. Dave finished before Adam Dave > Adam
2. Eileen finished after Betty Betty > Eileen
3. Adam finished before Charlie Adam > Charlie
4. Eileen finished after Dave Dave > Eileen
5. Betty finished before Adam Betty > Adam
6. Dave finished after Betty Betty > Dave
7. Charlie finished before Eileen Charlie > Eileen
From 1 and 3:
Dave > Adam > Charlie
Dave > Adam > Charlie > Eileen
Betty > Dave > Adam > Charlie > Eileen
Which is the final placing for all of the players, and can double checked with the remaining clues.