There were two major and related reasons why Diocletian wanted to persecute, and eventually get rid of, Christians in the Roman Empire.
First, this was a time when the Roman Empire was in decline. Diocletian felt, as many people have at various times in history, that the solution to contemporary problems was to return to past ways. For this reason, he wanted to return the Empire to the way it had been at its peak of power. One way to do this was to get rid of the Christians.
Relatedly, Diocletian felt that it was important that everyone in the Empire should perform the traditional religious rites that, the Roman pagans believed, appeased the gods. It is not clear whether Diocletian believed this out of religious conviction or if he saw it as a way to create more unity and cohesion in the Empire. One way or another, he wanted the traditional rites to be restored. Christians would not participate in those rites because doing so would imply that they acknowledged the existence of gods other than God.
For these reasons, Diocletian persecuted the Christians during his time as emperor.