Alexi Barov is one of the two "surviving" (if such can be said about one who is forsaken) descendants of the Barov family, along with Weldon Barov. As such, they are both heirs to the Barov family fortune. Although it is unknown how exactly they intend to deal with obtaining that fortune - since most of it is probably still buried somewhere within the treasuries of Caer Darrow - the brothers nonetheless strive for the four deeds that can be found somewhere in Scholomance. This inheritance has made them enemies, and Alexi will stop at nothing to become the only one surviving heir to the Barov fortune. He starts a line of quests that end in the killing of his brother, Weldon Barov on the Alliance side.
Alexi Barov may be a reference to Alexi/Alexei Stukov, a character from the Starcraft universe. His story is similar to Barov's; he was killed and later resurrected by the Zerg.