Really, I'd think that it might actually just be the oxidisation of a carbon chain that would make it, like most other plastic products are made. That's basically what they are, liquid plastic that bonds when in contact with blood, or other salt substances.
Well, actualy, proteines and peptides (the proteines smaller brothers) are polymers of amino acids. This is an amino acid:
http://en.wikipedia.org/wiki/Amino_acid. the left-hand side, with the N is the so called amino group and the right-hand side with the two O's is the acid side
The R is just a side group and there are 20 side groups used in living organisms.
Now you can tie amino acids together by having the acid group react with the amino group, which makes a bond from the C atom of the acid group to the N atom of the amino group, and you lose 2 H's and an O, to form a dipeptide. Repeat this and you get tripeptides, tetrapeptides etc.
The described reaction can be done chemicaly, but that's difficult. It's easier to have an organism do it for you, and that's usually done by genetically modifying a bacteria. That's for example how insulin for diabetics is made.
And for the record, one could argue that a peptide is a kind of bioplastic, but that's not a common definition. Would be apropriate for your hair though, since that's almost exclusively made of proteins.