Use of third-party inks might require you to download a small subroutine to turn off the 'counter' which tells you your cartridge(s) is/are empty.
Having said that, HP cartridges are about €16 for black and €18 for multicolour in Spain: third-part cartridges cost very little less and as has been said there's a risk of leakage or clogging associated with 'em. As for "refill your old cartridges yourself", don't even think about it.