“Every time no one’s around” or "every time when no one’s around"

The construction every time when... is not idiomatic. No examples of it can be found on Google Books Ngram Viewer.

Every time can either be followed by that or by the clause that follows without that. For example:

Every time that he comes here, he brings flowers

or

Every time he comes here, he brings flowers.

The construction no one's is likely to be understood as no one is although it might possibly be interpreted in some contexts as no one was or no one has.

So you need to phrase your sentences either in the present tense as:

I can't help but stare at him every time (that) no one's [is] around.

or, in the past tense as:

I couldn't help but stare at him every time (that) no one was around.

https://books.google.com/ngrams/graph?content=everry+time+when%2Cevery+time+tha

These both sound wrong; "no one's" is a contraction for "no one is", but we would want "no one was" there, instead.

That being said, the second one is better because the "when" is redundant - we already have said "time", so saying "when" is repeating ourselves a bit.

Also I would prefer "any" to "every", but the differences are subtle. Also, "anytime we were alone" is worth considering.

They are both correct.

The second form is a bit better, because it is shorter.

The meaning of both is exactly the same.