Next of Kin
You have to kill all members of the Draconis family. First you have to go to Applewatch, west of Bruma.
Talk to Perennia Draconis - she'll take you for a gift deliverer of some sort and provide you with a list of her children. Kill her and start your Draconis-killing marathon throughout Cyrodiil.
Matthias Draconis works in Umbacano Manor In Talos Plaza. You can ambush him any time of the day - just don't do it on the streets or the guards will be all over you.
Caelia Draconis is a city guard in Leyawiin. Each day around noon she's spending some time alone in a room in Three Sisters. If you can't sneak-kill her, she'll probably give you a standard city guard speech - you'll have to choose "resist arrest" option and finish her off.
Andreas Draconis is an innkeeper of the Drunken Dragon Inn in the eastern parts of Cyrodiil. His inn is always guarded by a fully armed guard, but you should manage to deal with the problem quietly (preferably during the night hours).
Sybilla Draconis is in Muck Valley Cavern, inhabited also by bears and wolves. It's quite simple here - go through the cave and do your thing.
The reward (500 Gold) and the next orders will be waiting for you in a well in Skingrad Castle Courtyard
Your next victim is a Khajiit named J'Ghasta, living in Bruma. Break into his house when the sun is down and enter the secret training room, using the Trap Door under some cloths in the basement. Defeat him (he's a master of unarmed combat) and go to Old Bridge, south of Imperial City. There's 500 Gold waiting for you in a crate below it, together with new orders.
Another typical execution. Go to The Flooded Mine, where you'll find (some swimming will be in order) an Argonian named Shaleez. Kill her and go to Fort Redman, south of Leyawiin, to get orders and 500 Gold from the coffin by the entrance.
A Matter of Honor
Alval Uvani is your next target. He's a traveling merchant, whom you can meet in various cities around Cyrodiil. His schedule was included with the orders. During the week, he's staying in Bravil, Skingrad and Bruma - and spends the seventh day in his home in Leyawiin.
You can kill him the old way - and the wisest would be to attack him on his way from one city to another. The problem is, he's a really powerful battlemage and it won't be easy to confront him openly. It's far easier to deal with him by planting some Mead for him to drink. He's allergic to honey and will become a sitting duck when he has a go of it.
Bringing this plan into morion won't be easy, though. To make him drink some mead, you'll have to steal ALL the food from a tavern he's staying at (so he can't actually choose anything else to food or drink), and then drop some mead here and there. He'll eventually drink some - killing him then won't be any problem.
Your reward is in a hollow stump behind the Stonewall Shield shop in Market District, Imperial City.