Billy had known Eva since they were children, and had developed feelings for her.
In London 1929 he tried to rekindle the relationship between him and Eva, but she wasn't reciprocating those feelings and was interested in Dorian instead. Both men had taken an instant dislike to each other, but while Dorian and Eva simple ignored Billy, he tried to make her see Dorian was a bad influence, claiming he'd heard stories.
One evening an earthquake struck, many people dying that night. Billy tried to convince Eva that Dorian was behind it, insisting that Dorian's recent arrival and the deaths weren't a coincidence. Billy soon found Dorian over the dead body of Casper Bonham, an infamously sadistic gangster, and jumped to the conclusion Dorian was the killer before re-entering the club.
That night Billy killed Eva, staging it to look like the other suffocation deaths. When Dorian found her body he found Billy, knocked him out and took him to the church of St Giles. There Billy was tied up as bait for the gaseous creature really responsible for the deaths, who killed him because of the suffering he'd inflicted upon Eva.
Billy was jealous and petty, ready to accuse Dorian of anything bad in the hopes of convincing Eva to come back to him. He probably killed her because she refused to change her mind. He and Dorian did not get on, Billy seeing Dorian as an opponent for Eva's affections while Dorian was content to ignore him. Billy was useless in a fight, as Dorian easily knocked him out.