Is there a way to see how much CPU usage per core a process is using?
Process Explorer at least can show you the threads of a given process and how much CPU those are using. Maybe you can get the core/cpu on which the threads are running somehow and then just add up. I have not much WinAPI experience, though. So maybe processor affinity can only be queried on processes.
You can't directly get the per-processer cycle times for a given process, but you can estimate it over time with QueryIdleProcessorCycleTime (as it returns idle times for all logical processors, i.e., cores) and QueryProcessCycleTime and assuming that it is the only non-idling process running.