There are several websites and services that allow you to view your runs on the web. I use SmashRun.com and Garmin Connect. Both have free account options and more features if you pay (or buy Garmin products).
That's step one. Pick a service/website and set up an account.
Step two is getting the data off of your iPhone. I use RunGap (free app, but paying $2 every 3 months gives you more export options), but HealthFit also works (one time $4 fee). Using one of these apps will allow you to export your run from the stock Activity app to an outside service.
Viola!
EDIT - Sorry. I misunderstood your question. There are no apps that I know of that allow you to view your runs on the AW. Even 3rd party running apps show very little past run history on the AW. Most just show a recent history of time/distance that summarizes. Very little detail.