To enter specific locations in Link's Awakening, you're going to have to walk around Koholint Island. It's not all easy, so you're going to have to wander around the game and explore many places to find ways to progress. One item you will need to progress through the game is the Bird Key. Here's how you find it in the game.
The position of the bird key in Link's Awakening
Before you can locate the Key to the Bird, you'll need the song The Song of the Frog's Soul. You will play this on your Ocarina to acquire the key. To find the song you are looking for, you will need to visit the Catfish's Maw dungeon. In addition, you will need the grappling hook and 300 rupees in reserve on you. Go Ukuku Prairie, and use the grappling hook to cross the chasm, to Signpost Maze. When you are there, read each sign to progress through the maze. Make sure you follow the signs exactly as you move through the maze.
Eventually you will find a staircase leading to an underground chamber and you will meet Mamu. He is willing to teach you the song, but only for 300 rupees. Pay and you will acquire the song.
With the song you want to go back to Mabe village and go into the weather vein. Push the weather vein aside, and go to the chasm hidden below. You will play the song to the skull to resurrect the rooster. He will follow you when you leave. You can wage war with him and you want to go to Dampé's hut north of your current location. You are looking for the path blocked by two rocks that you can now lift. Lift them up and throw them with your rooster in tow.
You don't want to go to the first ladder of the trail. Instead, you want the second set. Go up the ladder and remove the rocks at the top, then enter the cave. You will now enter the Tal Tal mountain range. Go through the tunnel and go up the mountain. When you come back to the stream, outside the cave, go right and continue until you are in the water using your fins. You will see a ladder, you can go up and take your rooster. You want to go inside the first cave entrance you come across.
The interior of the cave is relatively simple. The first rock room you enter, you want to go to the northern entrance. Push the rocks until you can use your grapple to cross. In the next room, you will need the help of the rooster to overcome this huge void. Approach him and he should let you grab his foot to fly around the corner to safety.
On the other side of the platform, you need to locate the Bird key. You are good to proceed to the eagle tower.