The first thing you need to do is speak to the Mistress of Admissions in Umbra. Her name is Aernya and she is usually wondering around outside of the Bloodletters.

She will tell you to take the form to the Mayor of Luna and get his signature. His name is Gorrow and his office seems to be at the Luna bank. You can find him wandering around there.

He is not the most honest man around because even though he knows you are no student of Aernya's he will offer to sign your form anyway if you will bring him some stuff for his stew. He wants you to bring him one unicorn rib, two ki-rin brains and five pixie legs.
At this point you may wanna make sure you are not close to negative karma because the best place I have found to get the pixie legs is the champ spawn near the Spirituality Shrine in Ilshenar. Go through the gate in Luna and then run north-east until you start running into the pixies. Start killing them and looting their bodies for their poor little legs.

When you have five of them it is time to go hunting the unicorns and the ki-rin. I have marked a rune and placed it in the runebook for this part of the quest on the front porch of my home which can be found standing all by itself in a clearing south-east of Luna at the edge of the Abyss.

By the way, all of the creatures you need to kill to get the ingredients Gorrow wants are good aligned creatures and you will lose karma for killing them. Now, after getting the stuff Gorrow wants mark them as quest items and return to him at Luna bank. He will sign your form and you will be off to Aernya again. Return to her with the signed form and you will be told to take it into the dungeon to have one of the instructors sign it.

Fell free to do so if you wish but enter the dungeon alone at your own peril. You have already done all you need to gain access to Bedlam and need do nothing further if you wish. At this point, as long as you have the signed tuition reimbursement form in your pack you can enter and exit the dungeon at will.
There is a champ spawn in the back of the dungeon as well as the named monsters in the main part of Bedlam that you will need to kill in order to gather the keys need to do the peerless located there.
And one more thing.....BEWARE OF RED DEATH....that mule has a kick!