Sometimes German and English grammar are not that different. Your German sentence
Das ist die Partitur des ersten Chopins Klavierkonzerts.
is correct, even though it does not parse very smoothly, and would be rendered thus in English:
This is the score of the first Chopin's piano concerto.
So we would be talking about the piano concerto of the first Chopin. Not what you meant.
Also, your English sentence
This is the score of Chopin's first piano concerto.
can be translated quite directly into German using the kind of pure genitive that you were trying to use:
Dies ist die Partitur Chopins ersten Klavierkonzerts.
However, that sounds a bit stilted, so one would usually use von for one of the two genitives:
Dies ist die Partitur von Chopins erstem Klavierkonzert.
Dies ist die Partitur des ersten Klavierkonzerts von Chopin.
As remarked by joelsa, the latter word order also again allows for
Dies ist die Partitur des ersten Klavierkonzerts Chopins.