just finished reading the manga, and based on what I've read, it seems that Mikasa and Levi are relatives.
Levi is the son of Kuchel Ackerman, a prostitute. Kenny Ackerman, Kuchel's older brother, comes to the city to see her; only to find that Kuchel is dead.
In one of her conversation with the group, she once told that their family lived in forest where their parents meet. Her father was hunted because of his surname "Ackerman", maybe they are hunted by Reiss family since Ackerman's are the only bloodline that cannot be brainwashed.
And Levi also asked Mikasa if she felt that kind of urge to fight. First was when Mikasa was kidnapped and rescued by Eren then Mikasa killed the third kidnapper. And for Levi, when Isabel and Farlan died, he was able to fight 5 titans (in manga but only one in OVA).