How did the Persian Empire overthrew the Babylonian Empire?

Persia did not have an empire before it took over the Babylonian Empire.

It was a subsidiary kingdom to the Medes. Under King Cyrus the Great, took Media over, and this combined force was able to defeat the Babylonians and absorb them, which was the beginning of the Persian Empire.

After that the Persians moved on into Asia Minor the rest of the Middle East, Central Asia and under King Cambyses took Egypt and Libya.