This mission always struck me as a bit odd for its simplicity. Your target is a Khajiiti named J'Ghasta. Aside from the fact that he offended a nobleman by refusing to marry a daughter whose dowry wasn't big enough, he has nothing in his background to warrant the attentions of a Speaker of the Black Hand. It's a simple assassination. But considering that the Cheydinhal Sanctuary is completely depopulated, perhaps it's just a temporary manpower shortage. And orders are orders, after all.
J'Ghasta lives in Bruma and apparently spends his free time practicing his Hand-to-Hand skills, which are considerable, and he seems to know that you're coming. On the bright side, you're not obligated to fight fairly (you're an assassin, remember?), so feel free to use anything that suits your fancy. And to make things a bit easier, he has paid off the town guards to look the other way in case your fight should spill outside. J'Ghasta may not be immediately visible in his house, so you'll need to search for the trapdoor to his basement training area. Once you've found him, take care of business.
Your payment for this job and instructions for the next are in a box under Old Bridge, which is across Lake Rumare, south of the Waterfront District.