I am undecided on the question of whether Strong was homosexual or heterosexual, really. Certainly something about Brandt snapped him into a place where the sexuality itself became much less important than the expression of power that it represented, and self-hating attraction is certainly something that could have that effect. But I'm not sure it's what it was, exactly.
One theory I rather like is that Strong, being rather screwed up himself pretty clearly, had some experience with rape or abuse in his past that has been stewing inside his brain looking for an outlet. When approached by the initial "victim," his jaws closed down around the case as an opportunity to revenge this past wrong on somebody who got off on the expression of power in his sexual experiences, somebody who could easily be seen as a surrogate for whatever rapist Strong was hoping to get his revenge on. When the case fell through, he couldn't let go, and attempted to get his revenge anyway by raping others of Brandt's ladies, hoping he could get Brandt the easy way by slapping him into jail. But that didn't work. Eventually, the obsession with the revenge became an obsession with the idea of expressing power over Brandt, which he expressed through the rape and murder of the women, in two different ways.
First, he used them as a surrogate for Brandt, raping them with short hair and his cologne, probably calling them by his name as he beat them. But he acknowledged that they were surrogates for Brandt, not Brandt himself, and kept the hair as personal evidence that he was more than Brandt. He went further than Brandt ever would in his lust for sexual power; he took it all the way, and with women that Brandt himself had failed to finish. He was more than the man he was revenging, better at what he perceived as their shared, disgusting perversions and that made him powerful enough to finish the job. He knew all along he would kill himself once it was complet: he hated that he liked the power trip, as he hated that Brandt liked it. But he had to bring Brandt down in the only way he could before he put himself out of his misery.