Evil Eagle is made up of an Eagle, as well as the Grim Creeper, the mini-boss found within Eagle's Tower. The fight taking place in the top of the tower. If Link falls off, he will have to restart the entire fight.
At the beginning of the fight, Link must equip the Hookshot and Mirror Shield. Evil Eagle's weak point is his beak. To make things better, he will fly across the screen and give Link free shots at his beak. Each time he flies by, Link must attack with the Hookshot.
His only real attack is a barrage of feathers and wind. When he begins this attack, Link must guard with the Mirror Shield and constantly walk towards Evil Eagle to avoid being blown off the tower. If Link can successfully keep from falling, the fight will not last long.
There is a much, much easier way. If you go to the next temple without defeating Evil Eagle, you can acquire the Magic Rod. Then, return to face Evil Eagle again. After four hits with the Magic Rod, the boss is deafeated.