Alexei was abusive — it’s not something made perfectly clear in the comics, it’s kept vague and briefly alluded to. Natalia carried on her secret relationship with James, the ones that the higher ups didn’t approve of. They wanted her with Alexei, the man considered to be the best Russia had to offer, the prime candidate. She didn’t love Alexei nearly as much as she love James, but she tried — brainwashing may have had something to do with it.
He might have been abusive because of the brain conditioning the Soviets put him under, or he might have been abusive because he sensed that Natalia didn’t love him as much as he thought she should.
It ended when Alexei was declared dead in a flight accident, and Natalia was put back in the Red Room; the experiment of their beloved weapon living a normal life called to a conclusion.
She’s not even sure her marriage to Alexei was real — the man is real enough. But she’s not sure if the idea of a loving (sometimes abusive) marriage was implanted, or if it was something she briefly experienced in some attempt to give the implanted memories more reason to stick.
But she knows that some part of her loves him, some part of her misses him, despite the evil he’s done — because she’s done just as much. They both have blood on their hands, and aside from James, he’s the only other person who truly understands.