Claire's first husband, Frank, is an Oxford history professor with a fascination for genealogy, so it's safe to say that he might be a little jealous to learn that Claire gets to meet his great-great-great-great (we've lost count) grandfather, Jonathan Randall, in the flesh. Although as far as ancestors go, Randall's pretty disappointing on account of how rape is one of his favorite past-times.
Frank and Claire love each other, or at least they really seem to in the two chapters we get to know Frank. They're fun and flirty and have lots—and we do mean lots—of sex. Claire tolerates Frank's long ramblings about genealogy, but though he's verbose, we have to wonder whether or not Frank is keeping secrets from her. Claire wonders what happened during the six years of war they were separated, but she never gets a chance to ask him before falling two hundred years into the past.
With no idea about what Frank is doing after Claire disappears, we're left to just postulate. Perhaps he also moves on with his life and remarries like she does? Your guess is as good as ours.