How did Jesus fulfill the moral law?

As a human person Jesus kept the moral law perfectly because He was also JHVH in the flesh, and He had no sin nature though He was tempted in all respects as we are.

Hebrews 4:15For we have not an high priest which cannot be touched with the feeling of our infirmities; but was in all points tempted like as we are, yet without sin.

If you mean did He abolish the moral law by keeping it, then the answer is no. The moral law stands today the same as it always has. The presumption that the moral law was abolished is a common misunderstanding. It's based on the following verse.

Romans 6:14For sin shall not have dominion over you: for ye are not under the law, but under grace.

To show definitively that the moral law still stands in spite of the commonly misunderstood Romans 6:14 requires a more sophisticated study than can be presented here in one or two sentences.