Start: Talk to the Guild Master of the Champion's Guild about Rune Plate Mail.
Requirements: Able to defeat a level 83 Dragon, and you need to be a member of the Champion's Guild. See the "Fighting the Dragon" section below for my level reccomendations.
What You Need: A Wizard's Mind Bomb, Silk, Lobster Pot, an Unfired Bowl, 12,000gp (or 2,000 if you're level 31 Magic), a Dragonfire Shield, 3 Planks, 90 Steel Nails and a Hammer.

Getting Started

Firstly, speak to the Guild Master of the Champion's Guild, found just southwest of Varrock, and ask him about Rune Plate Mail. He says that a friend of his named Oziach may have some for you, and he can be found just west of Edgeville, but you need to go around the stone wall to get to his house.

Ask Oziach if he will sell you some Rune Plate Mail, and then tell him that the Guild Master told you about him. Then ask how you can prove that you're a hero, and he says that he wants you to kill Elvarg the Dragon. Ask if you need anything, and he'll tell you that you need a Dragonfire Shield. Ask where you can find the dragon, and he tells you that it can be found on Crandor Isle, but the map has been torn into 3 pieces. He also gives you a Maze Key.

The Shield

Head to Lumbridge and talk to Duke Horacio in the castle about a shield to protect against dragons. He'll give you a Dragonfire Shield that you can use.

Map Piece 1

The first map piece can be found inside of Melzar's Maze, located just northwest of Rimmington and southwest of Falador. Use the Maze Key with the door to get inside, and then kill the Giant Rats until you find a Red Key. Use this with the door in the northwest corner and go up the ladder. Now kill the Ghosts until you find an Orange Key and use it with the door second from the north on the east wall, heading up a ladder in the small room after the door. Now kill the Skeletons until you get a Yellow Key which can be used on the southwest door. Go down the next 3 ladders and kill the Zombies to get a Blue Key. Use this with the blue painted door and kill the level 43 Melzar the Mad to get a Magenta Key. Use this with the next door and then kill the level 82 Lesser Demon to get a Green Key. Use it on the last door, open and search the chest for the Map Piece.

Map Piece 2

Visit the Oracle on Ice Mountain, located west of the Monastery. The mountain is also south of the Black Knight's Fortess. He says that you need several items in order to enter a room with the map piece inside. After obtaining a Wizard's Mind Bomb, piece of Silk, Lobster Pot and an Unfired Bowl, head down into the Dwarven Mines from the ladder to the south.

Head south past the Mining Store near the entrance, and then east the the room in the northeast corner of the mine with a chest inside. Open and search the chest inside for the second Map Piece.

Hint: A Wizard's Mind Bomb can be bought from the Falador pub for 2gp, and silk can be bought from either the Silk Seller in Al Kharid, or the Clothing Shop in Varrock for 30gp. A Lobster Pot can be bought for 20gp at the Port Sarim Fishing Shop, and an Unfired Bowl can be crafted, without baking it in an oven.

Map Piece 3

Now head to the Port Sarim Jail where you will find Wormbrain the Goblin. You can attack him with Magic or Range at him, but you'll need to be at least level 33 Magic so that you can Telekinetic Grab the map (or have 31 Magic and use a Mind Bomb). Otherwise, you'll need to talk to him and buy the piece for the not-so-cheap price of 10,000gp.

The Boat

Now that you've got the map pieces and Dragonfire Shield, you just need a boat and someone to take you to Crandor Isle. Go to Port Sarim and talk to Klarense, the owner of the Lady Lumbridge. He can be found east of the Grocery Store, and ask if you can buy the ship from him. He'll sell it to you for 2,000gp, and after buying it board the ship and climb below the deck. There's a large hole in the ship, which should be easily fixed.

You'll need 3 Planks and 90 Steel Nails to fix the hole in the ship. Nails can be smithed using Steel Bars, but you'll need to find the planks. Planks can be found in level 19 Wilderness, northeast of the Chaos Temple, which is north of Varrock. Bring the Planks, Nails and a hammer to the hole, and use each of the planks with it to fix the hole.

Getting to Crandor

Put the map together by using one piece with another to get a Crandor Map. Talk to Ned in Draynor Village, and ask if he will take you to Crandor Isle. He'll take your map and head off to the boat, so meet him below the deck. Talk to him again, tell him you're ready to go and the boat takes off to Crandor. It gets shipwrecked, but everyone is alright.

Now head north around the Island on the inside (not on the coastline), until you pass the King Scorpions. Now head southwest along the coast, and then head southeast past the Hobgoblins. Climb down the opening and head south to the gate.

Before even going inside, continue allong the corridor to the south, past the Skeletons and Lesser Demons, until you come to what appears to be a dead end. You can push the wall to open up the passageway, leading into the Karamja Volcano!

Should you happen to die (or you must teleport) before killing Elvarg, just take the boat from Port Sarim to Karamja, climb down into the Volcano and enter through the passage to the northwest to return here.

Fighting the Dragon

Now it's time to fight Elvarg the Dragon, which can be a tough battle for anyone. I'd suggest you to be at least level 45-50, but the battle will be long and very difficult. No matter what level you are, bring some decent food such as Lobsters or Swordfish, since the Dragon can hit for over 10 damage. And don't forget your Dragonfire Shield! I'd also suggest you to either use the Stab or Crush style of fighting, since Slash doesn't work very well.

After defeating the dragon you'll be teleported outside of the room and the quest will be complete. Congratulations!

Rewards

• 18,650 experience to your unused attack styles; Agressive: Attack & Defense experience; Accurate: Defense & Strength experience; Defensive: Attack & Strength experience 
• Ability to wear a Rune Plate Mail and Dragon Leather.
• 2 Quest Points