Receive a Slayer Task
The only successful way to level up is to receive a task from a slayer master and hunt down the specific monsters given in the task. If you hunt down and kill monsters NOT in the given slayer task, you will not be able to level up as quickly.
Slayer Masters
There are a total of seven Slayer Masters that can be sought out for a slayer task, beginning in Burthorpe with Turael or Spria. Each slayer master gives a task involving a specific monster. Once you receive the task, depart and hunt down the monster or the number of monsters assigned. Keep in mind that if you receive a task you do not like or do not want to do, you can request a new task from the first slayer master.
Slayer Monster Locations
Although monsters can be found almost anywhere, there are four locations where they are most likely to be found.
Slayer Tower -- located north of the city Canifis
Fremennik Slayer Dungeon -- a dungeon located in the Fremennik province
Chaos Tunnels -- a dungeon with five entrances located north of Edgeville and Varrock
Kuradal's Dungeon -- a dungeon located in the Ancient Tavern.
You can reach most of these locations by utilizing various fairies to drop you closer to the area.
Killing Zombies
While killing zombies seems to be the easiest way to level up, it should be remembered that you should only hunt the monsters assigned in a slayer task. Zombies tend to come at the end of the first two slayer tasks, therefore being worth more points. So in order to reach these monsters, it is best to receive the task from the appropriate master and proceed as quickly as possible.