Summarize:

You can get here by heading North from the Barbarian Village, which is just across the River Lum, right past the West Entrance of Varrock. Read all of the plaques on the jail cell doors, and then talk to the nearby Security Guard to gain access to the Training Center, which is up the steps right next to him. Talk to the Professor once you are done with the test. He should give you two Experience Reward Lamps upon completion of this test. There should be a poster which you can pull back to gain access to the Stronghold Dungeon. Go through it and ignore the muggers and Cockroach monsters. Climb up these stairs to get to the upper floor. Go to the left side of the room and pull the lever to unlock the door in the previous room you were in.
Go to the Misthalin Training Centre of Excellence. Enter the basement of this building and you should now be in a jail. Talk to the Professor in the Training Centre and complete the written test he gives you. Go back down to the jail and locate an open cell. Head to the Northeast and locate a set of stairs. Go through the door you just unlocked by going back down the stairs and heading through the door.