I have two slots on the back of my printer, one says line and the other says 2 ext. I have the black line coming from the printer to the the phone jack in the wall, another line coming from my phone jack to the slot saying line on my computer and the last line coming from my phone going to my phone slot on my computer. I cannot not get a dial tone from my computer do I have my printer hooked up right.
As far as I can tell from your query you have the hp plugged into the phone line correctly (presumably so you can use it as a fax). In order to use the telephone you need to plug the phone either into a Y adapter at the wall socket (you'll see why it's called a Y adapter when you buy one -- it turns one socket into two).
You could, perhaps, instead plug the telephone into the ext socket on the hp.
Do NOT plug the telephone into the computer (I have no idea what socket you are trying to plug it into -- just don't).
Now read the manual for your hp and figure out how to connect it to the computer -- presumably the printer socket or USB.