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You can start this one by going into the Blue Stripes' rented quarters. They rented out the whole house just to the left of the inn. You can find it by looking for the soldier watching the stairs (and possibly Thaler's agent, depending on your previous save). Just walk inside and into the back room to find Thirteen, Fenn and Ves sitting around the table. I suggest you save first just to be sure that you actually benefit from this quest.
If you talk to anyone, they'll mention that they had a bit of fun with the statue of a local god. Once they settle down, they'll push you into a game of knife throwing with Ves.
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Knife Throwing with Ves
This game isn't hard once you understand it. You'll be given some throwing daggers. There should be an aiming reticule on the screen. If there isn't, then reload the save that you made. It's a real pain to accurately throw without it. It's worth trying for the win, since it looks like you'll get some nice throwing knives as your reward.
The way that this will work is that the soldiers will call out targets for you and Ves to hit. If you hit all three, you'll win. If you take too long to throw, or miss, then you'll lose. I believe that it's the same three for Geralt, although I've seen two different orders for them. There's the oar to the left, which is in the corner. The cured ham hanging from the ceiling in the center and the map on the far right. I've seen it go map, ham and oar, and oar, ham and map.
Hitting the map is easy. Just face it and get a knife somewhere on it. The ham is the hardest. Aim just a little bit above it and you should be able to get a hit. The oar is tricky, but you should be able to aim at the top of it and get a hit on the flat side of it. Again, aim a little high.
Once that's done, some townfolks will come in and threaten the Blue Stripes. Roche will step up and pick a fight with them. You can choose to just let him go, or choose to intervene. If you intervene, you'll need to beat the persuasion check, or you'll end up in a fistfight with him.
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When that's done, you'll drink a bit, and then wind up in an arm wrestling match against Thirteen and Fenn. If you beat both of them, you'll get a fairly nice sword. The Sword of Creyden is a good enough sword for the beginning of the chapter, and it has a fairly nice buff to bleeding. It's worth a shot. They don't seem to be particularly tough opponents. If you've faced other arm wrestlers in the town, then this should be a breeze.
As always, just try to hold it early, and then edge your way to the right. If you can ride the bar over to the right, it will be an easy victory.
Once you beat both of them, the drinking will continue until you wind up half naked on Flotsam's beach.
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When you wake up and regain control, you'll be able to talk to a local man. You can either intimidate him, or pay him about 20 orens to get him to point you toward the madame. Go to the basement of the inn and back into the brothel area to find Margot. She'll either be in her room or in the general area.
Talk to her, and offer her a bribe of about 30 orens to get her to point you towards Ves. It seems like she was sober enough to try and stop you, so she should remember what happened. Go back to the headquarters and talk to Ves in back. If you ask about what happens, she'll recount the details and give you the key to the chest by the window. It will have all of your inventory inside. Take the chance to reequip now.
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Removing the Tattoo
The only remaining issue, of course, is your tattoo. If you don't like it, then you'll need to talk to Triss at the inn. Once she stops laughing, she'll explain that she can make a removal potion if you can bring her some white myrtle petals, wolf's aloe and green mold.
You probably have some white myrtle petals from the forest. It looks like wolf's aloe and green mold can only be bought though. Go to Anezka's house in Lobinden, and purchase them both from her. It won't be cheap, so bring at least 150 orens, or something that you can trade.
With the ingredients in hand, go back to Triss and ask about tattoo removal again. After she delivers her speech again, Geralt will be able to mention that he has the ingredients. You can then confirm that you want it removed. It'll disappear immediately.
If you miss the chance to remove it in Chapter 1, you'll get another chance in Chapter 2 with whichever sorceress you side with.
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All information and screenshots from a "Normal" difficulty playthrough of Witcher 2
Due to the branching nature of the game, your experience may differ. Feel free to note such differences below.