Is there any way to monitor kernel-mode code execution on Windows with a program like Process Monitor or is that something for WinDbg only?
The closest thing I'm aware of is DbgKit from Andrey Bazhan. It is not exactly what you are asking for, but it does provide a nice GUI extension on top of WinDbg (which can be used for kernel debugging). It's more like Process Explorer than Procmon.
What do you mean by monitoring kernel-mode code execution? There's a lot of things happening in kernel mode. For example file and registry operations will still be visible in ProcessMonitor, due to kernel code most of the time is executed in context of the user mode processes, i.e. outside of System process. Yet if you remove a filter on System process in ProcessMonitor, you'll be able to see kernel activity from System process too. Also there are tools like IrpMon or DriverMon which allow you to see IRP traffic/the calls to IRP_MJ_xx functions.