@OP, It is not practical to make a peptide bond from two free amino acids.* Instead, the Carboxylic acid is made into a derivative. Regarding the biochemical synthesis of peptide, what is done is to make what an organic chemist might call an active ester, an ester with a hydroxyl group of transfer RNA.
*I suppose that one could draw out the mechanism for acid-catalyzed or base-promoted hydrolysis of a peptide bond. The mechanism of formation would be the reverse, but the equilibrium favors hydrolysis.