Provided that I had financial backing and good search programmers…
If I were going to build a search engine on the sly:
I would launch a search engine. At first it would be a metasearch engine powered by other search feeds, but the feeds would be disguised, perhaps blended so you didn’t know it was a metasearch. I would add my own elements to the search serps like Wikipedia. While this is going on, I would be working on a crawler, one that spiders deep. I’d also be looking at ranking schemes and reverse engineering why my search providers rank sites the way they do. Not trying to violate patents, just figuring out why one site is more important than another. I would develop my own feed and test it. Once I felt I had a good quality algo, I would start seriously building a spidered search engine index. I would slowly start introducing my own feed into the serps. If it tested well with users, then the third party feed would become more backfill. The backfill would receed more as my index grew. No obvious change in the serps. Eventually you would quietly drop the meta search and use 100% your own feed.
Something like that. It would be a long game, building and adding features, brand and users over years.