Chances are that you can stuff a typical mATX motherboard into the HP case. However, as dalauder points out you may run into a few problem areas.
HP, like many OEM builders, use proprietary front panel connectors often. You may have to seperate the LED (HDD and Power), power switch, and reset switch wires to physically connect to the new motherboard.
Also, HP, Dell, and others often use rather weak PSUs in many of their systems. The spcific PSU you would be recycling may not be up to the task of powering a lot of GPUs. We would need to know the output rating of your current PSU to know what it can handle or not.
You may also need to look closely at your case's airflow. Adding new stuff into a case designed for less heat could create some problems for you. If the case will allow you to install both front and rear fans, you should be fine. Otherwise, be prudent in how much horsepower (generates heat, draws power) you decide to install.
Last, it will still look like an HP. Not much you can do there. But who cares?
Have fun!