share a printer over through the internet

I have a h p deskjet printer have had it couple of months has ink in it cant get it to print?

You could try cleaning the nozzles, and / or doing a printer head clean. Instructions on how to do this can be found on either the company's website, your manual, or the setup CD. You may have to do this more than once.
Whatever you do, DO NOT remove the ink carts and then replace them, unless you have brand new ones with which you are going to replace them with. Manufactures recommend that you keep the carts in place until you need to change them, and recommend you clean the nozzles / heads first before trying anything else.
You should use your printer at least once a week to keep the flow of ink in order, even if you just print out a test page. This stops the ink drying out on the print heads, which in turn causes a blockage. Your printer will then not print, even though you have full ink cartridges and a fully working printer.
Also, if this is a newly installed printer, check to make sure you are not trying to print on the old printer --- uninstall your old printer and reset the new one as your default printer.
Failing all else, there may well be something else wrong with the printer itself, which you will have to either return to the shop or contact the manufacturer.

How to share a printer over through the internet to connect the another printer?

The cheapest way to do this do this is by using MS Remote Desktop. Remote Desktop can take some time to configure and secure but it is free and will allow you to "forward" the printers from the client computer to the host computer. This will allow to connect to a computer at work from your home and run applications on the computer at work and print to your printer at home. The work computer will have to have the drivers installed for the home printer.

The hitch with any outside connection is with the company you work for and what their policies are. Outside connections would have to ported through the router and if your company has an IT department then they would have to have approval from management to open this connection.