The pump on my DeepCool AIO seems to be dead. How can I check to see if it is working?

There are numerous ways to check if the pump is functional. The two most common way is by feeling for vibrations, or listening for sounds from the pump. I would check the BIOS and check for an RPM reading for the fan header that you plugged the pump into. There is also numerous monitoring software out there that can be of great use, like CPUID and AIDA64.