Claude is one of the main enforcers for the Bee Hive Gangsters and stands out as someone who's actually onto the fake relationship between Raku and Chitoge—which puts him at odds with most other characters who buy into it. He's obsessed with making sure Raku doesn't mess with Chitoge, stalking her pretty much constantly to keep tabs on things.
His overprotectiveness has basically defined Chitoge's entire life. She's mentioned that growing up, she couldn't make friends because Claude was always running background checks on people and generally being overbearing about who she spent time with. It's the kind of thing that clearly frustrated her, though it probably says something about how serious he takes his job.
Beyond that, Claude raised Tsugumi to be a hitman for the organization. The running joke is that he apparently never realized Tsugumi is a girl—he even gave her a boy's name, Seshirou, which makes you wonder how oblivious he can actually be about the most obvious details.
Content compiled by AnimeList.moe from publicly available sources.