Skyrim:In the Shadows

Quick Walkthrough
#Read the Steward's Note.
#Take the ledger copy at Arcadia's Cauldron.
#Read the Business Ledger Copy.
#Investigate the Hall of the Dead.
#Check the stables.
#Investigate the Silent Moons Camp.
#Find the Assassin on the upper floor of Dragonsreach when the Jarl is unoccupied.
#Retrieve the Bow of Shadows.
#Speak to the Jarl's Steward.

Detailed Walkthrough
An Assassination Plot
Talk to Proventus Avenicci (or Brill if Balgruuf has been exiled) and ask if he has anything for you. He will hand you a note, explaining that he is concerned of an assassination plot on Jarl Balgruuf the Greater from an anonymous informer. He dare not tell it to you out loud due to the Jarl already having plenty of his plate, as well as the fact that he is unsure of its credibility.

Proventus would like you to visit Arcadia's Cauldron and look through their business ledger to spot any suspicious purchases; to this end he has asked Arcadia to leave you a copy to peruse.

Suspicious Purchases
Head to Arcadia's Cauldron and you will find the business ledger on the counter. Unlike the typical business ledgers, this one you can pick up; do so and look through it to find that two people - a Priest of Arkay and a Nord Stable Boy - have made purchases of poisons. Time to investigate and look at their motives for doing so.

The Hall of the Dead
Enter the Hall of the Dead and then the Whiterun Catacombs; note that if you have not entered this location before there will be hostile skeletons in this area - either kill or avoid them. Once you enter, descend down the flight of stairs to your left and you will find a dead skeever, a block of cheese, the poison in question and a note, explaining that the poison had been applied to the cheese in an attempt to rid the place of the skeever infestation.

Whiterun Stables
Exit Whiterun proper and down the path to the stables, where you will find a dead Nord behind a haystack of the leftmost stall. Beside him will be a lunar steel weapon. Loot the Nord and find upon him a note as well as a spell tome of Detect Life; like before, read the note and find that the Nord has been posing as a stable boy. The note also states that he was supposed to meet a Dark Elf at this location instead of at Silent Moons Camp and the spell tome was to help the Nord in locating him as the Dark Elf snuck up on him the last time, though he was apparently unable to get the spell working.

Off to Silent Moons Camp
Now that you have your next destination, make your way to Silent Moons Camp. Your objective is at the very back, so if you are so inclined you may opt to go around and hop over the wall, but whichever way you choose, a journal of an assassin will be lying on a table just outside the room containing the Lunar Forge. Again, pick it up and read it, finding largely ramblings but an important detail: the author of this book shall be waiting on the second floor of Dragonsreach, waiting for an opportune time to strike and kill the Jarl.

Back Where We Started
Return to Dragonsreach and head up the stairs on the right of the main hall to the second floor, then turn around and make your way out onto the balcony. The area looks clear, but once you reach the end you will find the Dark Elf assassin in question, who had been hiding under a cloak of invisibility; at this point he will become hostile and attack you. (You may also cast Detect Life to locate him if you have learned it from the spell tome you should have obtained earlier, however as he will cease to be invisible once you get close it is not necessary.) He is quite decently armed and can pose a fatal threat if you're of lower level, but luckily you have allies who will help if you lure him downstairs.

Note: Rather uniquely, the assassin is marked as protected, which means you must land the killing blow.

The assassin will not appear if the Jarl is occupied, and will not appear if you haven't started the quest Bleak Falls Barrow . The Jarl is considered occupied when the following quest stages are active:
Bleak Falls Barrow - Stage 10: When Jarl Balgruuf is showing you to Farengar.
Dragon Rising - Stages 10, 12, 13, 15, 20, 110, 120, 130, 150: During the scenes inside Dragonsreach at the beginning of the quest and after the Greybeards summon you to High Hrothgar.
The Fallen - Stages 25, 27 - When persuading the Jarl about to trap Odahviing in Dragonsreach.
At any point during Battle for Whiterun, Battle for Solitude, or Battle for Windhelm.

Once the assassin is dead, loot his corpse and find the Bow of Shadows. Speak to Proventus (or Brill) again and gain a monetary reward.

Quest Stages