The hobo merchant will not sell you the goth clothing just yet, that comes later along the main quest, on the second day after the announcement of the the new Taco Bell at the South Park mall, Cartman will tell you that the elves stole the stick in the middle of the night, and that the humans will need reinforcements from the goth kids.
Deliver Cartman's message to them, and they say they'll only help you if you look and act like them;
- To look like them, you'll need the goth equipment from the hobo merchant by the U-Stor-It, which he should now sell. (Allegedly, you only need the Goth Cap, Goth Clothes, and Goth Gloves equipment. Any goth hairstyles, glasses and makeup are supposedly optional. But, you might as well go for the full look.)
- Coffee - Head to the Tweek Bros. Coffee Shop and buy some Tweek Bros. Dark Roast. It is the only coffee dark enough to be considered goth. As a bonus, this will also make you friends with Tweek's parents.
- Cigarettes - There'll be some tough looking kids in the alley beteewn Jimbo's Guns and the bar. Beat 'em up and take their smokes.
At some point during this process, the elves will kidnap you and take you to their hideout, behind Kyle's House. They'll try to convince you to join with them, not Cartman, and recruit the goth kids for the elves. They'll let you go, you can decide who to ally with later (And there's a few friendship opportunities here).
Return to the goth kids wearing your new goth equipment, and they'll send you on another quest - to get a photo of yourself with an anti-conformity sign in front of the PTA. Various bits of story occur, and Randy will teach you a new spell, and sends you to investigate the Taco Bell before he'll let you take your picture. This leg of the quest will require some combat, so switch out of your goth gear. When you return with evidence against the Taco Bell, Randy will take your picture and send it to you (Talk with the PTA milling about after for another 5 friends).
Re-equip the goth gear and show the goth kids the evidence of you conforming to their non-conformity (If they keep telling you to get that picture at this point, you do not have your goth disguise fully equipped, I got stuck for a while here because I had some other black glove equipped). This isn't quite enough to sway them however, you must now show them you can dance like a goth. You must satisfactorily complete a brief DDR sequence - NOTE: Myself and a number of other PC players got a little stuck here, the DDR sequence is not controlled with WASD, but with the arrow keys. No amount of trying to use WASD or trying to out-think the game by not conforming (not pushing any buttons) will get you past this.
Now sufficiently convinced of your non-conformist ways, all four goth kids will friend you. Now you must decide who will you report their allegiance to.