Can you refill the black ink cartridge on the c7250? If so, how.

Check the new cartridge. Make sure you remove ALL the plastic protective strips that came with the
cartridge.

You cannot print unless both cartridges are installed and have a decent supply of ink left in them. It doesn't matter if your printing only black and wont need to use the color cartridge. It still have to be installed with ink for your printer to work. There is no way of getting around this problem.
I wish I had better news for you.

Using the HP Print Diagnostic Utility in WindowsUse the HP Print Diagnostic Utility to help you identify and fix common problems when printing.

First of all, I hate to say this but, be sure you are using a genuine HP 02 black cartridge. Refilled / re-manufactured cartridges can cause damage to the printheads because their formulas are sometimes too acidic.1) Press the SETUP button on the front panel by the display. 2) press the DOWN arrow until "Tools" is highlighted on the display and then press the OK button 3) Arrow down again until you see "Clean Printheads" and press OK again. 3) After cleaning, press Setup again.. Arrow down to "Print Report" and press OK. 4) Arrow down to "Self-test Report and hit OK again. Your printer will print out another page. If the black ink is flowing on the self-test page but yet not when u are using the computer, then there's got to be some software problem on the computer itslelf. If it still doesn't print from the black printhead the we usually have either a print head failure or an IDS (Ink Delivery System) failure.

I have to ask this question of you, just to make sure this is not an issue. Are you using a factory new ink or toner cartridge?
Please consider the following:
If you are using a refilled (the absolute worst possible choice) or third party cartridges (2nd. Worst possible choice) that may well be your problem. Most manufacturers build logic into their products to prevent the use of “Other Equipment Manufactured” cartridges as well as re-filled cartridges.
They do this to prevent problems from contaminated inks or toner causing problems with print quality or damaging the physical printhead or photo conductors.
If you are using factory new ink or toner cartridges, then please call the manufacturer’s Technical Support for assistance with the warranty on the cartridges
Don

Are you using a factory new ink cartridge?
Please consider the following:
If you are using a refilled or third party cartridge that may well be your problem. Most manufacturers build logic into their products to prevent the use of “Other Equipment Manufactured” cartridges as well as re-filled cartridges.
They do this to prevent problems from contaminated inks causing problems with print quality or damaging the physical printhead.
If you are using factory new cartridges, then please call the manufacturers Technical Support for assistance with the warranty on the cartridges
Don

The ink cartridge is bad get a new one after a few times of refilling the ink cartridge they go bad. Hope this helps ya

Cheers
Johnthetr870

“Lexmark does not support the refilling of its ink cartridges. Refilling the ink cartridges can cause them to leak, thus clogging or even damaging the print head. Any damage to your printer caused by refilled ink cartridges may not be covered under warranty”. Please use genuine Lexmark cartridges for best results.

By using Colorbyte's Imageprint software and their 'Phatte Black' cartridge.... 
www.colorbyte.com
I have not purchased it myself quite yet but im good friends with an Epson R&D guy who tested their system and highly recommends it.

HP #74 Cartridge Refill Instructions
Black cartridge
Instructions
1. Put the cartridge on some paper towels in an area where you can tolerate a spill.
2. Peal off the top label.
3. Fill the syringe with 20ml. of black ink.
4. Very S-L-O-W-L-Y inject ink into any hole (they all go to the same sponge, HP just used the same color mold for the black ink cartridges).
5. When finished, replace the top label or leave it off (it will make no difference in the printing).
6. Replace the cartridge and run 1 - 3 cleaning cycles as per printer instructions.
Or print the following pages:
http://www.refillinstructions.com/black.htm for Black

For best results, leave cartridge in printer for 8 hours after refilling so the ink can saturate the entire sponge.
A Refill Kit should include Ink plus the following:
One 30cc Syringe