It was an...unusual marriage, to say the least.
After countless years of insisting they would never marry, especially to each other, they found themselves in the typical wedding motions: discussing the type of wedding, meeting her parents, Bruce fending off Selina's mother's flirtations, talking about ceremonies, and her father wants it traditional Catholic, Selina would rather just sign the damn paper and get it over with, Dick's designing her dress, dear god. Eventually, they were wed and on their honeymoon around the world.
Which was, of course, merely an excuse for new training and deeper studies into newly discovered criminology techniques.
At home, back in Gotham, things were very much the same as they ever were. Selina had her own apartment where she could be alone, though she did spend much more of her time at the Manor. Her cats travelled with her back and forth and roamed freely about the grounds of Wayne Manor (the cats being the largest part of the "yes, I'll marry you" bargain). While Titus was at first very skeptical of this new person and all her cats moving in, he soon learned that she was clearly Mama, as the loud cat said, after Selina wrestled him to the ground, pet him, and called him a good boy.
Life with the children was one of the stranger parts. Dick would pick on her and call her "Mom", which usually earned him a cuff on the back of the head. She was much more motherly to Jason and Steph, though she insisted she'd rather be seen as an older sister than a mother; "mother" made her feel old and panicked. She and Tim were polite to each other, not overly friendly, but neither held any animosity toward the other. Damian was a different story. After two weeks of finding her food poisoned (mildly, of course, or Bruce would have his hide), her clothing ruined, her possessions destroyed, and her equipment stolen, Selina sat him down for a talk.
"Look kid. You don't like me, and that's fine, because, frankly, I don't like you much, either. But I'm not having this. I'm not telling your father what you've been doing, because you're a big boy and can handle it yourself. I'm not trying to be your mother and I'm not going to try to be your mother, but dammit, I live in this house and you're going to have to accept that. Got it?"
"-tt- As if I would ever accept you into this household. You are my father's concubine and nothing more! The fact that you--"
"...I already have Cordelia and Lavinia. I don't need another--"
"I found the most adorable Siamese kitten the other night. Reminds me a bit of Lian, actually..."
That sorted the problem very quickly.
She and Bruce had their occasional domestic squabbles ("Selina, stop stealing the hostess' jewels", "Bruce, just hug your kids, dammit"), and their occasional loud, dramatic fights (usually watched by any and all present members of the household over the cameras). The biggest fight came after Bruce caught Selina breaking into a government building for secret files and all but said that since she was his wife, she couldn't keep being Catwoman, or he might have to turn her in. It was an empty threat, but one that Selina didn't take kindly to at all.
The next day, when Bruce came home from work, he found Selina dolled up like the stereotypical perfect '50s housewife, smiling and simpering as she doted on him. It took two hours and a minor breakdown over pearls, but he relented to let her stay Catwoman, as long as she only stole evidence for cases.
Overall, it was a fairly good marriage. They continued much as they had before their legal marriage, so they didn't kill each other with the transition, and they had at least three major separations a year, which only served to keep the relationship intact.
