You can't catch death fish with alley catfish. Normally, you can catch death fish with angelfish, which you can catch with alley catfish, but you don't necessarily need bait. You can fish without any bait, but it means that you'll catch less fish, and you'll catch random fish, the easiest fish being more numerous, so you'll catch more vampire fish and even more red herring. You'll catch a death fish only if you're very lucky, so be patient.
If you've got a fish tank, and populate it with 2 death fish, you can breed them. You can have up to 6 fish at a time of identical quality ; once I have 6 of them, I take 4 to my inventory so the 2 remaining ones can breed again. Or you can feed omniplants with death fish too. You can even clone fish with the science station. So you don't necessarily have to catch them.