Answer: Betty won, then Dave, Adam, Charles 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 Charles Adam > Charles
4. Eileen finished after Dave Dave > Eileen
5. Betty finished before Adam Betty > Adam
6. Dave finished after Betty Betty > Dave
7. Charles finished before Eileen Charles > Eileen
From 1 and 3:
Dave > Adam > Charles
Dave > Adam > Charles > Eileen
Betty > Dave > Adam > Charles > Eileen
Which is the final placing for all of the players, and can be double checked with the remaining clues.