This may seem daunting and/or expensive, but don't worry! Despite what the achievement says, you (currently) don't have to watch 25 different movies. You can start, stop and resume the same movie 25 times and it will unlock.
When I signed up for Vudu it came with 10 free movies to own. Currently, the signup deal is 5 free HDX movies when you join, so with luck you should already own at least one movie.
1. Go the the "My Vudu" twist on the main page.
2. Click on one of the movies you own to go to its details page.
3. Click on the top left button on that page (Either "Watch SD" or "Watch HDX").
4. Once the movie starts playing, back out.
5. Go back to step 3 and repeat 24 more times.
The achievement should pop as soon as the video starts playing for the twenty-fifth time.
Along the way you'll have gotten this as well: