In my experience when an ESXi host locks up there's not usually a lot of checking to do. About 99% of the host lock-ups I've seen have involved a PSOD - the error will be displayed on the console of the host (in the text on the PSOD) but at that point filesystems are no longer working and it won't be recorded in any log. If you have persistent logs they might have some details on what happened just before the lock-up, which may or may not be related. If you are booting ESXi off of a small device (thumb-drive, SD-card, etc. I believe anything under 5GB but not sure on that number) there is NO persistent log setup by default.
In the end, the majority of the issues I've seen ended up being related to known issues in NIC/HBA drivers/firmware (usually Emulex) and are resolved after the updates are applied.