During each Crucible match, you will steadily earn experience points while playing the game. The more experience points you earn during the match, the stronger your Crucible Hunter becomes, making you a stronger force while to play against your opponents, and to help your teammates to progress through the game.
You can see your Hunter’s level on the lower left-hand section of the user interface. You will need to take out additional enemies and players to level up. You will gain additional levels for taking out a player, versus taking an enemy NPC wandering around the map.
You start the game at level one with an array of abilities. Each level you will gain additional health, and your abilities will steadily become stronger and do more damage. Right now, there is no maximum character level in the game. If two teams continue to duke it out, not gaining objectives or not progressively doing better than the other, then players on both sides could consistently become more powerful. The character level might change in the future, depending on balancing changes to Crucible.
Everyone can play Crucible for free through Steam. The game is a PC-exclusive, so you should not wait out for it to the PlayStation 4 or Xbox One.