I mentioned this in a comment once before and never made it into a full feature-request.
Having been a moderator for half a year now and handled quite a bit of comment flags, I often run across some... not-so-great comments that just make you think to yourself "Why in the world would you ever say that?" Of course they get deleted, but that doesn't really do anything other than removing the comment.
Unlike with answers, deleted comments aren't visible to the person who posted them. There's no indication to the user that the comment was a bad comment, unless they happen to remember they posted it and realize it got deleted. There's no indication they did something wrong. It just vanishes.
There's no real way for a moderator to easily see that the user is a consistent problem. Sure, they could go look through their comment history and look for the red backgrounds, but most of the time a moderator has no reason to actually do that. I'm not going to investigate every single user whose comment I delete for rudeness to see if it's a pattern - that's way too time-consuming.
While comments may be second-class citizens, they are often the very first replies that new users (or perhaps any user) encounters after posting a question. They're the first impression. That's not something that we should be taking lightly.
So, I suggest that some sort of automatic flag raised by Community, to indicate that the user has had x comments deleted by a moderator within the past y days (note, I don't think self-deleted comments should be counted). This would give moderators a quick flag that tells them this user has had a particularly bad run of rude or unuseful comments and perhaps needs a little message in their inbox.
It's not something that's particularly hard to investigate once we actually know about it, but the current system just allows users to continue on posting these comments because we never really know about it outside of the single instance.