What kind of marriage does Winston have?

Winston's marriage was one in which there was evidently no love, at least on Winston's part. He recalls that his wife was basically uninterested in sex and submitted to it only because it was "our duty to the party"—that is, in order to have children. Katherine's personality also seems, in his recollections, to have been dull in the extreme.
We are hearing this only from Winston, of course—just as the entire novel, though written in the third person, is told from Winston's point of view. It's not entirely clear if Orwell is presenting the information about Winston's failed marriage as unreliable, distorted by Winston's bitterness about the whole situation in which he finds himself and the world under totalitarian rule. Winston describes Katherine to Julia as "a wife I can't get rid of." Julia, as a woman who not only enjoys sex but (despite Winston's judgement of her as not very intellectual) bonds with him on a personal level, is the antithesis of Katherine. It's ironic that Winston sees uninhibited sexuality, such as Julia's, as a force that can destroy the Party. The totalitarian rule of the Party, like the actual Soviet Union of Orwell's time, is atheistic, and yet it has a conservative attitude about sexual freedom and marriage. Katherine evidently viewed procreation—which she and Winston did not achieve—as the only purpose of marriage, and this was partly, or mostly, the reason that Winston's marriage was a failure. The memory of it still torments him, though he and his wife are permanently separated.

Winston Smith is still technically married to Katherine, although they've been separated for years. He never loved Katherine, and he only really married her out of duty. He thought her vulgar, stupid, and unthinkingly loyal to the Party. But in Oceania, people don't marry for love; they marry for the purposes of procreation and nothing else. And this is where Winston and Katherine's marriage started to hit the rocks. Despite "doing their duty," (a euphemism for having sex), they were unable to have children. It was Katherine's inability to conceive that led directly to the couple's separation. It was just as well that they did split up, though; not only did Winston not love Katherine, he actually hated her and once came close to pushing her off a cliff.
As Winston and Katherine never officially got divorced, Winston's subsequent sexual liaisons with Julia are adulterous and therefore constitute a serious crime against the state.