Champion levels (CL) are an advancement system that starts once you hit level 50, and talk to the king in your realm's capital city. You can gain up to 15 champion levels. Each level grants additional hit points, power and a champion point you can spend on new abilities.
When you first hit level 50, talk to the King in the Kings Throne Room of your capital city to unlock champion levels. You start off at Champion Level 0. If you left click click on the label to the left of your XP bar (in the character status summary window, by default in the lower right half of your screen), you can change it from displaying "Exp" to "Chp." When you mouse over the Chp text it will say "earning X champion level XP" where X is the champion level you are working towards unlocking. For example when you first talk to the King it will say "Earning Champion Level 1 XP." When you earn new champion levels you need to talk to the King to activate whatever levels you have unlocked. You can complete several Champion levels at once if you have earned more than one when you talk to the King.
Earning Champion Levels
There are several ways to gain CL xp. Killing any player or any mob that is high enough level grants CL xp. Champion levels 1-10 all take exactly the same amount of CL xp to earn, levels 11-15 each take significantly more. Items that grant bonus XP like Orion's Belt also grant bonus CL xp. For example, it's well worth carrying around Orion's Belt and putting it on before you use CL xp scrolls or turn in the final glass quests. There are two basic routes to quickly earning champion levels (Update: One method as of patch 1.125).
Bountypoint Quests and Champion xp scrolls 
There are both daily and weekly quests that will reward significant CL xp and bounty points, which can be used to purchase the Champion XP scrolls from the bounty point NPC. The details of them are a bit beyond the scope of this page, and have been subject to frequent developer changes in any case. Goto the bounty point page HERE for more information.
On Ywain a popular bounty quest is the "Supplies for the Cause" granted by the Cargomaster NPCs stationed at specific docks in the new Frontiers. For Hibernia see the respective realms boat dock located near Dun Crauchon in Emain (Hib), Cear Beno in Hadrian's Wall (Alb) or Bledmeer Faste in Odin's Gate (Mid). Completing this quest yields 19,200 CL xp which is approximately half a level of champion xp below CL10. It also rewards approx 11,000 bounty points (if veteran not EC) as long as you have the buggane bonus and are sub RR4L0. Note: The buggane bonus is +75% bonus until 3L9. You can pick up the quest on Gaheris, but currently it can't be completed on that server.
To further speed up getting champion levels, you can use any bounty points you earn to buy the experience scrolls from the Bounty point NPC in each realms border town. Look for the bounty master, they are found in the main border relic towns and inside the Main Cities of each realm. The easiest one to find is a Bounty Master standing near the same quest givers that used to grant Scouting Ellan Vannin. These merchants sell scrolls of "Bounty Master's Insight/Greater Insight/Champion Insight" These scrolls cost 50/500/10,000 bounty points and grant CLxp one bub for the Greater insight and 10 bubs of CLxp if you are for CL1-10. After the Champion insight will reward less and less CL XP due to the higher CL xp requirements for each respective level. Repeating these bounty quests and using all the points to buy scrolls, you can burn through all of your champion levels in a few hours.
NOTE: CL 1-10 will cost approx 100,000 bountypoints and CL 10-15 will cost approx 100,000 bountypoints. (updated as of patch 1.126)