I'm not sure if this normal but I find I can rub off layers upon layers of "dead" skin (on my face) after soaking in a warm bath for 15-20 minutes. Once my skin feels soft I'll cleanse then rinse, and while my skin is wet I'll use gentle circular motions with the tips of my fingers to remove all this loose/dead skin. Afterwards my skin feels smooth but it's shiny which i heard was a sign of over exfoliation. I don't use any scrubs or any products with gritty bits in them.
I have very oily skin which I thought I should mention as it could be related.
I've started to use Stridex 2% Salicylic acid each night as chemical exfoliation is more gentle, but it seems i still need to remove this dead skin after a few days by manually rubbing otherwise my skin looks dull and starts to flake (I do moisturise morning and night).
So is this buildup normal or is it a sign that I'm doing something wrong?
This is weird because you say you have "very oily skin". Maybe when you shower/bathe, you could use slightly colder water. If it gets worse, stick to lukewarm water, never hot. Exfoliate only a few times per week, maybe 2 times per week since you seem have to be having alot of dead skin. After showering/bathing, moisturize. And every night, you could use aloe vera on your face, it moisturizes and should help healing. If it persists after a week or so, go to a dermatologist and get it checked out.
Good luck, you'll be fine! 
This is weird because you say you have "very oily skin". Maybe when you shower/bathe, you could use slightly colder water. If it gets worse, stick to lukewarm water, never hot. Exfoliate only a few times per week, maybe 2 times per week since you seem have to be having alot of dead skin. After showering/bathing, moisturize. And every night, you could use aloe vera on your face, it moisturizes and should help healing. If it persists after a week or so, go to a dermatologist and get it checked out.
Good luck, you'll be fine!
I've never really had hot steamy baths, but I could start to use lukewarm water instead. I'm wondering if there is anything wrong with using my fingertips to exfoliate as my face is a little red and blotchy afterwards but that goes away after a few minutes. I'm just a little unsure if its dead skin thats supposed to come off or if it's dry skin that needs a heavy moisturiser.