The quest that will give you this achievement is Up to the Challenge. This will take you approximately 3 days to complete as you only recieve 5 Aspirant's Seal per day. In order to begin working up in the ranks, you must first visit Justicar Mariel Trueheart. She's standing in the entrace of the big tent on the Western edge of the Tournament Grounds. She will send you to speak with Arcanist Taelis (Alliance) or Magister Edien Sunhollow (Horde) to sign up the for Tournament. Each day you will recieve a random daily and along with Learning the Reins and Training in the Field. Below is a list of the random dailies you may receive.

1. A Blade Fit For A Champion: Retrieve the Ashwood Brand. First travel to Grizzly Hills and head East to Ashwood Lake (60, 50). There will be a lot of blue Lake Frogs that you need to /kiss. It's important to use your Warts-B-Gone Lip Balm before you kiss them, every time. It may take a bit before the frog turns into the maiden, however, if someone else finds her you can also speak with her to complete the quest. Please remember that you need to actually speak with her and click on the speech bubble in order to receive the weapon.
2. The Edge of Winter: You are to retrieve Winter's Edge and bring it back to your respective quest giver. Lord Everblaze is who you need to kill for the ember. He's in Crystalsong Forest at (50, 74) up on a cliff. After you've retrieved the ember, you need to head out to Howling Fjord. Winter's Breath Lake is North at (42, 20). All you need to do is target the maiden encased in ice and use the ember in your inventory to free her.
3. A Worthy Weapon: Retrieve the Blade of Drak'Mar. First, you need to gather 4 Winter Hyacinths. They are on the North side of the Ironwall Dam in Icecrown. Afterwards, head out to Dragonblight's Drak'Mar Lake (93, 26). Simply use the Winter Hyacinths and the maiden should appear and spawn the blade.

Once you've acquired 15 Aspirant's Seals, you can complete Up to the Challenge and get your achievement!