Scheduled for Execution
It's payback time. Your target is Valen Dreth, the Dark Elf in the cell across from yours who insulted you at the start of the game. You are to take him out, but you'll get a bonus if you avoid killing any guards in the process. This means that you will have to make your way back through the sewers and into the Prison cell where you started. If you noticed, the key that Baurus gave you that let you into the sewers was removed from your inventory at that time and that sewer entrance has been locked again, as has the outer grate to the sewer exit. Not to worry. Valtieri will provide you with a key to get into the sewers again and the other locks can be worked around with a small supply of lockpicks or suitable spells.
If you ask around the guildhall, Teinaava will tell you that you should be careful because the sewers are home to some fearsome creatures. This might be true in other sections, but even leveled into the high teens I only encountered Rats, Mud Crabs and some Slaughterfish. Your major worry is going to be the Imperial Guards once you get back into the Sanctum, Imperial Subterrane and Prison. They are supposed to be dumbed down for this quest, but they still have an ungodly detection ability. So the toughest part is going to be managing to get through this without having to kill any of them. If you have a Paralyze spell (targeted or touch), use it for a quick get-away in case you're detected, but be prepared to run like hell. Whatever you do, once you enter the Imperial sections, don't use weapons on anyone other than Dreth or you run the risk of killing a guard and forfeiting your bonus.
If you'd like to do this the quick way, fortify yourself appropriately and go charging in the front door of the Prison, though the offices, and down to Dreth's cell. Kill him and go back out the way you came. You'll probably pick up a minor bounty for trespassing (perhaps for killing Dreth, too) and there is the problem of the guards stopping you, but it is possible to do it that way. Not nearly as much fun as the sneaky way, but it will get the job done. The long way is to use the sewers. The guards in the sewers will not stop you; they will behave like any other bad guy and just attack you. But remember that they are part of a "good guys" faction.
If you have not completed the "Ultimate Heist" quest for the Thieves Guild, this may be your first time trying to navigate the sewers on your own (you'll do it with Baurus as part of the Main Quest, but he's leading the way). The sewers have a lot of twists and turns and backtracking in order to move from section to section. You probably won't encounter anything truly dangerous, so just follow your compass and be cautious. When you reach the entrance to the Sanctum, you'll need to deal with a 3-pin lock in order to exit the sewers.
Once in the Sanctum, you'll get the opportunity to overhear a couple of guards talking. They're pretty much convinced that their entire presence here is a waste of time, but you can pick up a little office gossip in the process. Once they are done talking, one will go guard the entrance to the secret tunnel where the assassin who killed the emperor came from and the other will go guard the exit to the Imperial Subterrane. There is a guard captain in the room where the emperor was killed, but he doesn't move from his table and it's not very difficult to sneak in and pick his pocket for some useful keys. There is a chest in there with a 4-pin lock on it if you're of a mind to snag a little extra loot. The whole area is pretty dark, so it's not difficult to stay in the shadows and sneak in and out.
Once you're in the Imperial Subterrane, you'll really need to watch out for roaming guards. There are a couple of them in the area and this is where I always manage to get detected, even with a Chameleon ring and potion (which should stack) and 50+ Sneak. If you can avoid the guard, continue sneaking your way to the Prison where you'll come out in your old cell. Dreth is still in the cell directly across from you and you can listen to his exchange with the guard (assuming that there isn't a guard hot on your tail from the Subterrane), who will then leave and go up to the Prison offices.
Telaendril will suggest that you simply put an arrow into him through his cell door and gro-Balmog suggests hacking Dreth and anything else that gets in your way (his usual advice). If you picked up the keys from the guard captain in the Sanctum or use a bow, you won't need to deal with the lock on Dreth's cell. Either way works and your only remaining problem is getting back out again. If you are undetected up to this point, you might have better luck simply exiting throught the Prison offices. While you are technically trespassing, you can probably make it out without incurring a bounty if you just bolt for the door. If you're caught, cough up the 40 golds and you'll be deposited outside the Prison offices, which is where you'd be anyway (minus any stolen goodies you might have had in your possession).
Report back to Valtieri for your money. If you met the requirement of not killing any guards, he'll also give you the "Scales of Pitiless Justice" which boost your Strength, Intelligence and Agility by 2 points each, while dropping your Personality by 2 points while the Scales are in your inventory. They're a very nice addition to your magical goodies collection, but may become a problem when you have to do anything that requires Personality (like Mechants, persuasion or Illusion). You'll either need to remove them from your inventory or carry around Personality boosting stuff to temporarily offset the point loss.