There are few places that information is available, but since kernel is a multitasking beast one has to look in multiple places:
The simple one is the Console log "type kernel in the filter box", but not all activity is reported there since Kernel issues millions of instructions in a Milli-second.
Also the Activity Monitor is a good source of information.
You could use the ps -A > process.txt
to see all running processes.
You could use the sudo iosnoop
to see all disk activity.
You could use kextstat > ~/Desktop/kexts.txt
to see all loaded drivers.
In other words, it is too busy for you to follow and a continuous output file would not make senses.