You should post a phrase with context and show the entire sentence (or even the sentence before & after). I am going to guess that this is a pun on "clinically ill." Clinically ill means that someone has already been officially diagnosed by doctors to have a certain disease, usually to mean that this person officially has a mental illness. Therefore, "clinically chill" probably means that this person is really relaxed and laid-back. This phrase doesn't actually exist, but it sounds similar to clinically ill, so it's a nice pun.

Context would really help, but I’ll guess. You’ve understood “chill” correctly, so we need to understand “clinically”. “Clinically” can be used as an intensifier in describing someone’s physical or emotional state — the idea is that a doctor could evaluate this person, perhaps run some medical tests, and then diagnose the person as “chill”. Since the doctor declared this person “chill” in a clinical setting (by direct observation), this person can be considered “clinically chill”. Of course, this is just meant as a funny figure of speech, as nobody actually goes to a doctor for such a diagnosis. A more common expression is “clinically depressed”, meaning a person is not just feeling blue, but that he has been medically diagnosed with depression. Edited for a typo.

Not sure what to make of this one. More context might help. "Clinically" is sometimes used facetiously to suggest that an expert medical opinion would support one'sn observation. This person doesn't just seem chill.: you could probably take them to the local hospital for testing and the doctors would confirm that they are indeed"chill". So you are comically treating it like a scientific conclusion. You are right that "clinically" is also used to mean "in a detached way", but I have a harder time figuring out how that could be used, even facetiously, with "chill". P.S. Love Richard Ayoade's work! So good as Moss...

You need to give the context. Otherwise can't help.

It's not really a phrase with any meaning.