Why did Solomon's soul belong to the Devil?

Short Answer

Solomon has implicitly sold his soul to the Devil through Common Law.

REAPER: Solomon.

SOLOMON: What... are you?

REAPER: I am the Devil's Reaper. I am here to claim you, Solomon Kane. Your deal is done.

SOLOMON: What deal? I made no deal.

REAPER: There was a deal. And your soul is the price. Your life of murder and greed is over.

SOLOMON: You cannot take my soul.

REAPER: Bow your head before me. The Devil will have his due.

Full Answer

The film is an origin story for the character Solomon Kane. Our first introduction is him acting as a mercenary, battling the Ottoman empire. As a general rule, mercenaries are typically perceived as violent, especially in the Middle Ages.

This violent version of Solomon Kane leads to the Devil attempting to reap his soul, after which he turns to fighting evil as a way to make amends for his past. Basically, the Constantine method of getting to Heaven.

One online description has this to say about the original stories:

Delving deeper, it should be noted that despite his good intentions, Solomon Kane is a tortured soul. While we never learn the reasons behind his obsession, it’s clear that Solomon Kane views battling the evil in the world as his personal cross to bear.

I could imagine trying to save his soul would give Kane a reason to bear that cross.

At the end when Solomon finds his father, his father says he summoned the sorcerer to heal Solomon's brother after his fall. Then he proceeds to say he begged the sorcerer and offered him "everything" which made me believe that Solomon's father offered his wealth, lands, etc and Solomon's soul.

Further, when Solomon speaks with Malachi after finding his father, Malachi says:

"Your father was a child, a pathetic fool who made a pact with the devil. He betrayed you. Your soul is damned."

So it does seem that it was Solomon's father who offered Solomon's soul the Devil.