For example, can I use it while I'm talking to a kid or could it sound vulgar? So when do you exactly use it?
There is a turn of phrase similar to "sick to hell" which is used by native speakers: "sick to death." As in:
I am sick to death of seeing this movie on TV.
The definition of sick here is not "ill" or "unwell," but rather:
filled with disgust or chagrin
The meaning overall is that you have had enough, you're fed up, you're tired of the same thing occurring repeatedly. "To death" is just an intensifier. I have also heard "sick to God" (same meaning).
If the meaning of "sick" you want is "unwell," then I don't think the expression "sick to hell" is too vulgar so much as it is too strange. It's not something said by native speakers. "Sick as hell" might be used -- either in the meaning of "unwell," or "fed up" as above -- but even in colloquial speech, formations like "really sick" and "super sick" and so on are much more common. You can use one of those when speaking to a child.
For some people, using hell as a descriptor (as opposed to referring to the place) is vulgar. It would probably be best to not say that around kids, unless you know for sure that their parents are okay with it.
An alternative word to use would be heck, which is much more acceptable. It's a euphemism, and euphemisms for swear words are generally okay around children.
Also, I believe the phrase you're looking for is "sick as hell", not "sick to hell".