I couldn't find a source that traced Robilar's progress room by room through the Tomb of Horrors, but according to the Dragonsfoot forum thread:
Robilar losing all of his orc flunkies at the onset, in the initial passage. When he found the tomb of the demi-lich Robilar scooped all the magical treasures he could into his bag of holding and ran off leaving the demi-lich hanging. [sic]
In other words, it seems Robilar took with him orc minions to activate the tomb's multitudinous traps, and he was careful to leave undisturbed anything suspicious, such as a creepy bejeweled skull.
Robilar's own account
On his site "Lake Geneva Original RPG Campaign" author and game designer Rob Kuntz, Robilar's player, was asked by user Thorsten S. about Robilar's adventures in the tomb. Kuntz was kind enough to reply in the site's comments, saying first that
I [Robilar] was careful and proceeded slowly; when I came upon the demi-lich I merely hefted the nearby treasure and ran out of the room, which seemed to perplex Gary. I'm not so sure of what more to tell as it happened very long ago, so the main points must suffice.…
Then, in another comment, Kuntz offers a few more details, saying
Gary called me and said he had a new level to playtest. We used to playtest these between us upon one of us finishing a level. Both of us always suspected the worst, as we had always thrown big curves at each other and our players over a year's worth of solid day-to-day playtests and play.
Five of the five orcs died at the entrance, one whom Robilar dispatched because it refused my order to advance into the entry corridor, the other four from the pits within said corridor.
I don't recall using this stratagem before; it's a left-over tactic from wargaming days.
The rest of it is vague. This was an initial playtest, so Gary may have changed parts for the final published form….
Kuntz concludes in another comment:
I do recall the face [that] I probed with a pole, the latter which came up an inch or two short afterwards. That and the pits… let me know that this was a trapped based adventure. Thus I slowed all processes of exploration accordingly—i.e., I was doubly suspicious of everything.
…As anyone should be when exploring the Tomb of Horrors!
Special thanks to Thorsten S. for his assistance in generating this answer.