This will be your first MAJOR contract. Our job is to head to Markarth and seek out the apothecary's assistant Muiri and form a contract with her. So head on over to Markarth and find Muiri. Once finding her, she will spill the whole story about the person she wants dead (whether you listen to it all or not is up to you).
The important thing is that she wants one Alain Dufont dead, who is the leader of a group of bandits. He's located in the dwarven ruins of Raldbthar. She will also offer you an OPTIONAL kill, one Nilsine Shatter-Shield, who lives in Windhelm. One thing at a time, let's head over to Raldbthar.
Chances are good that by now you've stormed a fort or two filled with bandits and the like (or mages, or some other group). This is pretty much exactly like that: everyone is hostile and you should go in guns blazing.
There's plenty to do here though, with chests to loot and gates to pick (you can even sneak around the mark and use ballista machines to kill him, if you can pick a certain gate). No matter how you do it, kill Alain and all his chums. Make sure you find the STEALTH SKILLBOOK, "2920, Last Seed, v8" while you're here, since it's next to the bandit campfire on a table.
Now as for the optional **** kill. Head to Windhelm. Nilsine belongs to CLAN Shatter-Shield, who owns a large house in the city. If you want to take her out at night (all stealthy like) like I did, you need to be somewhat decent at lockpicking, as Shatter-Shield manor has an Expert lock on it.
Once you get that though, it's all good. Make your way up to the second floor (the shrouded boots are great here) and you'll find her asleep on the right. Now get in a sneak attack to kill her. Loot the body and silently head back out into the night.
With both kills done; head back to Markarth and go find Muiri. She'll pay you 800 gold and give you Muiri's Ring , which makes created potions 15% more powerful. Now we'll have to go back and report our success to Astrid. This will complete the quest, and after some talking with Astrid we will get a new quest: Whispers in the Dark.