Meredith often disagrees with a majority of Alvin's decisions. Presumably, they are not divorced, and have an on and off relationship, constantly getting back together and separating.
Meredith and Alvin were 'separated' but not divorced. He was living in the garage while Meredith and Lexi lived in the house. They were having issues and Meredith didn't want Alvin in the house.
In Called in Dead, Alvin told Voight that his and Meredith's relationship had improved and he was going to move back into the house. He brings Meredith roses but stops to see her surrounded by gunmen who threaten to kill her because of a case that Intelligence has been working on. He desperately fights to save her and he eventually does and saves his shaken wife.
Alvin is back in the house but his and Meredith's relationship took a turn when Trudy told Alvin that someone had been trying to contact him. He found it was to do with another daughter he never knew he had. He keeps this a secret and tries to help Michelle out first but she's distant and doesn't care.
When Alvin eventually tells Meredith that he has a daughter from when he was undercover that he didn't know about, Meredith is calm and tells him it's okay, and that they can find Michelle a good foster home and be done with this. Alvin is surprised and tells her he can't just abandon his daughter. Meredith says Lexi is his daughter and this will destroy her, but Alvin refuses to let go. It does upset Lexi, who ends up crying and angry at him.
Alvin then orders a DNA test with the help of Antonio. He takes her mouthguard when she's finished boxing with Antonio and sends off the sample. However, he gets close to her and she thanks him for helping her and he doesn't have the heart to open the results, so he throws the envelope away. Meredith is very angered to hear this.
In Deathtrap, 51 responded to a warehouse fire. Alvin showed up and told Boden that Lexi was also inside and begs them to back in and look for her, which they do for him. They find Lexi and save her but when she's taken to hospital, her condition isn't good. In Emotional Proximity, Alvin is told by Rhodes that Lexi has too many third-degree burns and internal damage and that they can't save her. Alvin is distraught to hear this, and Meredith also rushes to the hospital to see her daughter. Alvin and Meredith hold Lexi's hands and hug her, comforting her soon before the line goes flat and she passes away.