The story was published in the mid 1960s, around 1967-68 in Analog or The Magazine of Fantasy & Science Fiction. I delivered both subscriptions to a young guy in Belfast when I was a paper boy and he would pass me his read copies when I delivered the new ones.
The character is bio-engineered, a bit like the Terminator, but with normal sentient self awareness, working as a kind of vigilante detective who can, because of extremely advanced biologically engineered senses, see the otherwise cloaked 'normal' aliens and kills them by whatever means possible.
Like the story Eight O’Clock in the Morning by Ray Nelson, first published in the November 1963 issue of The Magazine of Fantasy & Science Fiction. Which was later made into a film in 1988, They Live, by John Carpenter. He also has narrow escapes, when he crushes one against a wall with a car when he's ambushed.
He also escapes from them when he's trapped on a ship and has to swim a mile under water (with a stainless steel endoskeleton - go figure) to escape them. I thought it was The Stainless Steel Rat by Harry Harrison but apparently not...