You need to check to make sure the PCM controlled ground isn't shorted to ground before you throw a PCM at it (VERY COMMON). Otherwise you're just chucking parts and shooting in the dark. There are a few ways to skin this cat. But I believe this is the quickest way.
With an INCANDECENT test light connected to Battery Negative: With Key On/Engine Off, Unplug #5 injector. Gently touch Pin 1 (Orange/Dark Green Tracer). Test light should light.
With Key On/Engine Off, Switch Test light to Battery Positive. Gently touch Pin 2 (Pink/Black Tracer). Test light SHOULD NOT light.
If the test light illuminates by touching Pin 2 connected to battery positive with the key on/engine off, it can mean a couple of things. Most likely is that the control wire (injector ground) running back to the PCM is shorted to a permanent ground which means you're constantly dumping fuel into the cylinder because it's firing all the time. Most of the time the short is located behind the valve cover where the fuel injector wiring runs back to the PCM.
OR, it could mean that the driver for the #5 injector inside the PCM is shorted to ground (less likely but possible.)
If the test light doesn't illuminate by touching pin 2 while connected to battery positive, then you need to check PCM control by installing a Noid light with the engine running. It should flash. If the noid light doesn't flash, you need to run continuity from Pin 2 to PCM to verify the control wire isn't broken. If wire integrity is good, THEN you need a PCM.