"For this reason the Jews tried all the harder to kill him; not only was he breaking the Sabbath, but he was even calling God his own Father, making himself equal [ison] with God." - NIV.
The Jews were wrongly accusing Jesus of making himself equal to God in the authority to change the Law.
Jesus had just healed a man on the Sabbath. Jewish tradition had added numerous restrictions to the Sabbath law over the years.
Jesus would certainly not break the Law which was still in effect until his sacrificial death.
So, when the Jews accused him of making himself equal to God (by "breaking" the Sabbath traditions of the Jews and claiming that it was not wrong to heal on the Sabbath), they were insisting that their tradition of not healing on the Sabbath was actually God's will. And, therefore, Jesus' claim that it was lawful to do so was "making himself equal to God" (in this case of healing and "changing" the Law - which would be God's right alone).
It seems reasonably certain that the Jews didn't really believe Jesus was actually claiming to be God but attempting to usurp God's authority in this one respect. But, since these were Jesus' enemies who were making this false charge at John 5:18, it really matters very little what they claimed.
What does matter, however, is what Jesus claimed. How did Jesus answer this false charge by his enemies?
"To this charge Jesus replied, `In truth, in very truth I tell you, the Son can do nothing by himself; he does only what he sees the Father doing....'" - John 5:19, NEB.
So Jesus did not claim that he was Almighty God or even equal to Him. He clearly told the Jews that he was not God, but that, even as God's spokesman, he could not act upon his own initiative. Can we really picture the Almighty God of the universe saying that he could do nothing on his own initiative?
We find, then, that the Jews made a charge, and Jesus refuted it. He never claimed to be God. He never claimed to be equal to God.
For more, see:
John 5:18 ("Equal": Ison) (Examining the Trinity)
What is the meaning of John 5:18? (rs p. 209-p. 220; Watchtower Online Library)
Exposing the False Reasoning Behind Trinity Proof Texts (Examining the Trinity)