i've been working on improving my traveling skincare routine for lots of summer travel as well, and i never check a bag!
you'll want to use a quart sized plastic bag to hold your containers. if you have more skincare than will fit in one clear quart sized bag, use two, but make sure that each one is in a different bag (ie purse/backpack and suitcase), so it looks like it's for two sets of luggage.
i have quite a few steps in my skincare routine, and lots of them are liquidy and i've experienced leakage with previous cheap TSA-compliant squeeze bottles. i just bought this 5ml eyedropper bottle kit and plan to decant and label them for each liquid item so hopefully that leaks less. opaque ones would probably be more ideal due to the sun exposure protection, but oh well.
for stuff like conditioner and lotion, the usual TSA-compliant squeeze bottles will work great.
if i didn't have such sensitive skin, i would use all of the free skincare samples i have on vacation for sure, they're so small and convenient.