Black doesn't work on HP psc 1350 printer

Printer unplugged from power, and USB cable unplugged from computer. Open the printer again, and take all of the ink cartridges out. You may want to don a pair of dishwashing gloves, (or something similar), to keep any chance of ink getting on your fingers.

Use a clean, lint-free, cloth, and soak it in as hot a water as you can stand, then wring it out until it's damp.
Wipe the area of the inkjet ports on the body of the printer.
To explain where these tiny ports are located, let's go back to the ink cartridge itself.
Before you installed the ink cartridge, you had to remove a piece of cellophane tape, or aluminum foil.
This area on the ink cartridge, is where it's inkjet ports are located.
Where these inkjet ports are on the ink cartridge, and where they line up on the printer body, is where the corresponding inkjet ports are on the body of the printer.

I'm suggesting, that ink has dried in these inkjet ports for the black ink cartridge. The warm water cloth will help dissolve this ink, and clear the inkjets on the printer body.
I would wipe in one direction only, and turn the cloth to a clean area, as the dried ink builds up on it. When hardly any ink is present on the cloth, I would then use a hair dryer set to low. Move the hair dryer constantly, and do not hold it in one place.

You want this are dry before you reinstall the ink cartridge. Once you are satisfied it is dry, reinstall the ink cartridges, close the printer.
Plug the printer back into power. (Dry hands!)
Plug the USB cable back into the computer. Wait until the printer has rest itself back up, then turn press the Power On button.
(Some printers rest themselves when they are plugged back into power, some rest themselves only when the Power On button is pushed)

I would align the heads, and print out a test sheet.
You may have to go to Diagnostics, and have the printer check itself. (Start>HP>Photosmart>Help,
will guide you to the Diagnostics section)