So, we completed the first game week (which took two sessions of actual play time) and I now have 15 conquest (mostly from hero deaths). My avatar is Captain Bones and I chose the leviathan plot. I already have lawlessness and eldritch lore. What should I purchase next? A lieutenant (Siren?), a treachery upgrade (for crushing blow), an avatar upgrade or just save for a monster updrade?

These conquest tokens are burning a hole in my pocket...

Eldritch to silver, definitely. They're the most powerful group and they're cheap for you. From a min-max standpoint it probably should have been your first upgrade as it's the one that will pay for itself the fastest.

Because she's so cheap, I'd buy the Siren, then save for Eldritch upgrade.

Every week the OL does not siege something is, from the overland map perspective, a week lost.

Ispher said:

+1
To me, this is a change from RtL. The Map pressures are different, and you want to get the Siren into play ASAP (Also to harrass the heroes - a TPK now can completely screw them over) and the Siren is entirely safe unless the heroes have Thorn or the OL screws up badly. AFter that, saving for Eldritch silver upgrade is definitely the highest priority.
And trust me on this one. Exploding skeletons are a very important source of kills if the hero play defensively .

Siren, Silver Eldrich, Exploding Skeletons. That should suit you through the start of Copper, then I guess it's time for good old Crushing Blow.

Sorry, I didn't realize that the Siren was only 5xp to purchase. Everyone else is absolutely right about getting her first.