In order to unlock the Mighty Eagle in the first place, you need to fail a level 3 times in a row. Doesn't matter what one. Once you have done that, you can start using it. I would suggest launching all of your birds except the last one first to destroy as much as you can with them. Then quickly fire your Mighty Eagle after to destroy everything else. The only way to get 100% Eagle score on a level is to destroy everything on the map. Pigs... blocks... everything.
UPDATE: Since the latest patch, you can now launch your last bird and while it's in the air, use a Mighty Eagle. This gives you an extra bird to help those struggling on certain levels. Happy hunting!
If you are stuck on a level, here are guides for almost all of the levels:
Classic - http://www.angrybirdsnest.com/catego...-birds/page/6/
Seasons - http://www.angrybirdsnest.com/catego...eagle-seasons/
Rio - http://www.angrybirdsnest.com/catego...hty-eagle-rio/