This Seeking Tech guide will explain how you can get the Cleared: Metal Slug 3 trophy from the PlayStation 4 version of Metal Slug Anthology.
This will unlock when you beat all five levels in Metal Slug 3.
To make things easier, make sure that the Continue setting is set to Free Play, which should give you unlimited lives.
You can find the video playthrough, along with the time stamps, below:
0:29 Mission #1
6:02 Mission #2
14:22 Mission #3
23:32 Mission #4
32:24 Final Mission