I have a backstory for an Argonian character that I really like but I'm having trouble deciding what my character shoud be like, what quests he should do, etc.

Here's his backstory.
In Black Marsh, a new assassin order is rising. The Shadowscales are rising again, but they're not like the Shadowscales of old. Hatchlings born under The Shadow are still trained in assassination from birth, but they do not go to the DB. They go to the Shadowscales. My character is a member of this organization.

He was tasked with killing Ulfric Stormcloak but before he could put an arrow in his head, Ulfric was ambushed and in the chaos my character was captured. From there he was taken to Helgen and you know the rest.

Let's see, quests....

Seeing that the shadowscales do not go to the dark brotherhood, you should probably want to stay away from that.

Civil war:
FIrst, why do the shadowscales/argonia want Ulfric dead? And don't just say "cause the way he treats the dock workers," because that's nonsense. By which I mean they wouldn't send an assassin to assassinate ulfric just because he's a sub-racist who lets Shittershield underpay the dockworkers. There has to be some political reason behind it, because let's be frank: Argonia isn't going to care about some immigrant dock workers in another country anymore than the Mexican government is going to care about underpaid mexican kids working in america.
Honestly, nothing that I can see makes sense in the political context of killing ulfric for the args. At least, not right away.
If they want Ulfric dead so Argonia can mount its own invasion of Skyrim (OH THE IRONY!), then killing ulfric would be the last thing they should be doing. Literally. They should be helping ulfric kick out the imps from skyrim, just so they can immediately pull an Ulfric on Ulfic and take a still battle-weary skyrim completely by surprise before it has time to lick its wounds. An independant skyrim is an easier target than an imperial skyrim.
If they want to help the empire (um... why?) then yeah join the empire.
If they're trying to help the Thalmor (again... why? AFAIK Argonia doesn't give two shits about the Thalmor either way) again, killing ulfric would probably be the last thing they'd want to do, for which you should see Ulfric's Dossier. They should be trying to keep him alive for as long as possible to wear down both the empire and the skyrim locals.
If they're trying to combat the Thalmor (again, doesn't make sense because Argonia doesn't really give two shits about the Aldmeri Dominion.) Then... I suppose I can see this. Though even here they'd be better off tipping the scales in Ulfric's favor, so they can kill him and take Skyrim for themselves.

Other quests:
Thieves guild: Might be a good place to start. Haven't done it personally so I can't offer any advice.
Winterhold: Doubt your character is a mage, but if they are... sure why not?
Companions: eh I wouldn't.
Main quest: Side with Greybeards/Parthie. No shadowscale would side with the blades. Like, ever. In fact, once they stop being useful to you, you should do the Thalmor a favor and do the honor of killing the last surviving blades yourself.

3918142 I was planning to kill the blades anyway. As for Ulfric, basically it's like a DB contract. Someone wants him dead, weather it be the Emporer or someone who has a grudge against the Stormcloaks, so they asked The Shadowscales for Ulfric's head. And I'm probably gonna destroy the DB.