I highly reccomend reading "Devil in the White City".

In the book, the author described Holmes's method to ensure no one suspected anything.

With so many people in the city for the upcoming World's Fair, there was a glut of craftsmen who were looking to work on the fair, but setteled for anything. He would work the crews for a few weeks and then withhold pay. They would complain and argue, but ultimately leave disgruntled. Rinse and repeat.

He drew the ire of A LOT of companies where he bought raw materials and furniture. He kept them on the hook for as long as possible and used his apparently astonishing charisma to keep them at bay.

The only people who seemed to become suspicious were the crews who helped install the gas lines throughout the house. They noticed that he had weirdly placed spigots and dials into particular rooms, strange arrangements they had not seen before. Of course, no ever imagined that someone as proper and kind as H.H. Holmes could ever be up to any sort of evil.

Sort of the same case for the people who installed the incinerator in the basement. It was the quality that could disintegrate bodies, but they never raised issue with what his purpose with it might have been.

TL;DR: Refusing to pay crews and hiring new ones every few weeks. He kept his debtors creditors at bay through small payments and charisma. Apparently, people back then could not imagine that someone could be that twisted.

EDIT: I believe the schematics were his own, so everything ended up a little funny. I could be remembering that incorrectly though. I don't remember reading anything about official inspection that happened.

Couple of follow up questions, brought up by the answers of users which talked about how this question is covered in Devil in the White City:

Was there an architect involved or did Holmes design all of it alone (I suppose he'd have to)? Do we know how did he aquire knowledge needed for that? The World's Fair was a very confusing time, but would this also mean that a construction of gimmicky/unusual buildings outside the Fair was usual at the time? I imagine that even a worker brought there for one day might raise an eyebrow when asked to install a door leading to a brick wall. So, for these cases, do we know if Holmes had an answer for such enquiries? And lastly, was there any official inspection of standards that the building had to go through, before, during or after construction?

EDIT: Hiya, since this comment will probably stay on top till an answer according to the standards of the sub might be posted, I'll just ask anyone who has a tip for a podcast or a book, or knows a bit, but is not an expert, to definitely PM me, I'll be happy to read and check these things out. But we shouldn't make the comment section into an unnecessary graveyard.