To be honest, I would build a new Haswell system for a minuscule amount of money vs. adding a GPU to an HP proprietary motherboard. You will probably have a decent bottle neck in the PCI-e Lane, FSBus, RAM, etc... The R7 260x is a decent little card, and is much more advanced than the hardware you already have. If you're planning on getting a decent PSU and GPU, what's stopping you from getting a new i3, H-81 mobo, and some new RAM? You could build a good solid machine for about $400. Just my two cents.
Otherwise, as long as you have the PSU, and PCI-e slot to put the GPU in, it should work. You certainly won't get all the performance the card offers though.
Blessings,
pacdrum_88