Slaves could not become citizens while they were still enslaved. However, a slave who was freed beame a free person and a citizen.
Slaves in the Roman Empire could be freed by their masters. This could happen through the slave buying their freedom or through the master simply granting the slave's freedom. When this happened, the slave became a free person and a citizen.
Freed slaves were allowed to vote in Roman elections. However, they were not allowed to run for office and they could not (in the rare event that they could get rich enough) ascend to be part of the senatorial or equestrian orders. The children of freedmen were regular Roman citizens with no legal limitations on them.