Generally speaking, the rights of a convicted individual are protected by due process, which initially derived from England’s Magna Carta, 1215. Due process is covered in the Fifth and Fourteenth amendments. Each amendment essentially says the same thing about due process, but the Fifth applies only to protection of the individual from the Federal Government. The Fourteenth amendment includes protection of the individual against state governments as well. While there is no direct language to absolutely guarantee the right to appeal a criminal conviction, it was noted in a case, Frank v. Magnum that a conviction obtained via mob-domination is contrary to due process. Since this case (see link), it has been frequently stated that there should be a process in place to protect the right of the convicted to habeas corpus, due process and appeal. The absence of such a process has been deemed unconstitutional.
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.