The important thing to know about Spear of Vengeance is that she is more than just the ghost of the woman once called Kalista. The champion called Kalista is hundreds (maybe thousands) of souls in the shape of a single being. When someone pledges their soul to Kalista they join the pile of souls and lose their individuality. This swarms of souls is guided by the one thing each soul had in common: a desire for vengeance.
The soul of the original human Kalista is, I suppose, the hive queen. Hence the wraith Kalista looks and speaks like the human woman. Maybe human Kalista'a soul is the strongest, or maybe it's just because she was the first. But her in-game lines show that Kalista can only vaguely recall the human she once was.
Note: Yorick's spooky cape is a similar example of a swarm of souls that have merged together. Maybe the Maiden of the Mist is as well, but I'm uncertain.
So...
what does Kalista need souls for?
She doesn't need them for anything. Desperate people pledge their strength to her, and thus become a part of the swarm.
Is she cursed with the duty of accepting contracts, or does she do so willingly?
Willingly. The souls that make up Kalista no longer remember who they once where, but they remember that they seek vengeance. So Kalista brings vengeance wherever she, because it is her only desire.
What is she doing when she doesn't have any contracts?
I'm only speculating on this one.
The undead of the Shadow Isles can typically only exist within the Black Mist. I think when someone summons Kalista (like the woman from the story), they allow Kalista to leave the Shadow Isles and manifest outside of the Mist. When no one has summoned her, I imagine she goes back to brooding in the Shadow Isles, where the Mist is. She wants revenge on Hecarim, remember, so that will keep her busy.
In the event of a Harrowing (the Black Mist manifesting on mainland Valoran), Kalista is probably part of the carnage. But while Thresh and Hecarim will just kill anyone they find, Kalista hunts down those she considers guilty, wether there's a contract on them or not.
I hope my answers have been useful. :)