I browsed Dirichlets Werke today and was kind of surprised by two remarks that he made on p. 354 (Über die Bestimmung ...) and p. 372 (Sur l'usage ...). In the second paper, he claims (my translation)
I have applied these principles to a demonstration of the remarkable formula given by Legendre for expressing in an approximate manner how many prime numbers there are below an arbitrary, but very large, limit.
In a handwritten note on the reprint he sent to Gauss he remarked that $\sum 1/\log n$ (this is Gauss's version of the PNT, at least if you replace the sum by an integral) is a better estimate than Legendre's.
I am a little bit puzzled as to why Dirichlet's claim to have proved the prime number theorem is not discussed anywhere in the literature. Or is it?