There is no specific right to appeal a criminal case granted by the Constitution of the United States. The right to appeal a case may be inferred from a couple of places in the Constitution, but, as the West's Encyclopedia of American Law (see link below) says,
There is no absolute right of appeal for all decisions rendered by a lower court or administrative agency.
The right to appeal a case may be inferred from Article I, Section 8 of the Constitution. One clause of that section states that Congress has the right to set up courts that will be inferior to the Supreme Court. This implies that there will be appellate courts, which also implies that appeals will happen.
The right to appeal can also be inferred from Article III, Section 2. That section says (among other things) that the Supreme Court will have appellate jurisdiction in certain types of cases. This, too, implies that appeals are permitted.