After you complete Dungeon 4: Snake, you should be able to get 12 heart containers, which lets you get the Magical Sword. After that, you should go to Dungeon 5: Lizard.
Optional: Get Some Bombs
Bombs are necessary for progressing through the next dungeon. You should probably get some before you go. You can kill Blue Octoroks and Blue Moblins for a chance to get bombs.
How to Get to the Dungeon
Dungeon 5 is in the northern part of the map, a few screens east of center. To get there from where you started the game, go east, then north until you can't go farther north, then go east until you see water. Go up the stairs, then go west twice. Go up the north stairs. You will arrive in a screen that looks identical to the one that you just left. Keep going north until you reach the dungeon.
Get a Key
Go east from the entrance and kill the Pols Voices to get a Key.
Get a Second Key
Go west, then north. Use the Blue Candle to light up the room. There is a Gibdo in this room holding a key. Kill it and get the key.
Bomb the middle of the west wall and go through the hole. If you don't have bombs, you can try killing the rest of the Gibdos in this room and going north, then east, then north and killing the Gibdos there in case any of them drop bombs.
After going through the hole, you can kill one of the Gibdos in here to get more bombs.
Get the Whistle
Put a bomb in the middle of the west wall and go through the hole. Kill all the blue Darknuts, then push the leftmost block up or down, and go down the stairs.
Follow the path until you are upstairs again. Go west through the locked door.
Kill all the blue Darknuts and then push the block to make stairs appear. Go down those stairs and get the Whistle, also known as the Recorder. It will be useful later.
Get a Third Key
Go back upstairs, then go east. Go south through the locked door and kill the Keese to get a key.
Get a Bomb Upgrade
Bomb the east wall and go through the hole to find an Old Man who will increase your bomb carrying capacity if you pay him 100 rupees. If you do, you will be able to carry 12 bombs instead of 8.
Get a Fourth Key
Go back to the room with the stairs in the middle, push the leftmost block up or down, and go down the stairs. Follow the path until you are upstairs again. Go east twice, use the Blue Candle to light the room, then kill the Gibdos and go north.
If you want a third key, you can kill the Dodongos in this room and go west, then kill the Zols to get a key. Be sure to go through the east door afterward to return to the Dodongo room.
Optional: Get the Map
Unlock the north door in the Dodongo room and go through it to get the Map, then go south to return to the Dodongo room.
Optional: Get a Fourth Key
Go east from the Dodongo room, and use the Blue Candle to light up the room. You can optionally kill the Zols in here to get a blue rupee. Then go north. If you want another key, kill the Gibdos to get a key.
Optional: Get the Compass
Go north. Kill the red Darknuts if you want the Compass.
Get a Fifth Key
Go north to get a sixth key.
Optional: Get a Sixth Key
Go west through the locked door and kill the Gibdos to get another key.
Go to the Dungeon Boss
Go west, and kill or avoid the Pols Voices. Then unlock the west door to reach the dungeon boss.
The statues in the corners will spit fireballs throughout the battle. Digdogger is invincible, but use the Whistle to make Digdogger turn into a small Digdogger, also known as Digdogger Kid. The kid is vulnerable to the sword and other weapons.
Get the Triforce Piece
After you kill Digdogger, get the heart container, then go north to get the fifth Triforce Piece.