I am trying to cluster data. Each point in this dataset is connected to some other points. I want to define clusters "depending on how much the points are connected to each other". After some research, I read about k-core clustering (and saw it has applications in social networking for instance). I think this is the algorithm I want to apply, but, according to what I found, it can only be used to "visualize" a network. Isn't it possible to cluster data with this method?
I think you want to address the "cohesive subgroup" literature rather than "clustering." k-cores are part of the cohesive subgroup literature and it sounds like they're might be the measurement you want. If you read some of the early literature on them (e.g., Seidman  in Social Networks), you'll understand the theory behind them better. I have no idea why you arrived at the conclusion that they can only be used to "visualize" a network; your post is the first time I have ever heard that.
While k-cores might be the way to go, I would suggest reading more broadly into cohesive subgroups, as you might find cliques, k-cliques, cutpoints, etc more appropriate. To start reading into this literature, I would recommend the Chapter 7 in Wasserman and Faust (1994).