It is definitely correct, but as Matt said, you could make it flow better with any of his revisions. :)

This sentence is correct in English. It does, however, come off a little choppy and weird. I believe a native would say : "This play has broken a record, running 100 performances straight." "This play has broken the current record of consecutive performances, running 100 times." "This play has broken the record for a single play, running 100 consecutive performances."

That sentence is very awkward.. Maybe these would work better: This play is known for breaking the record of having 100 performances. This play has a record of 100 performances. This play has broken the record of 100 performances. The examples that Matt091 listed are also good. Mine are simplified examples :)