@lewm- in fairness, the very first response in this thread was the suggestion to read Neil Antin's work, which is available for free. Yes, it is voluminous but explains the chemistry and steps, the fairly minimal equipment needed and why the steps Neil outlines work. The total cost is far below the OP's budget.
I prefer to use a vacuum type rcm as part of my process-- I used a VPI 16.5 (which started life as a 16) back in the early '80s-- it was still operating fine when I gave it to a friend in 2017. I don't have experience with other wand type vacuums that may be available (or the clones of the Nitty Gritty that either don't have a motor or allow you to use a small household vacuum cleaner). Some here have suggested other equally inexpensive alternatives.
@clearthinker - I don't regard this as a fetish but a practical reality given that I buy mostly older copies. Often, the contamination comes from previous, "bad" cleanings-- some spray that might have been applied. Once I have effectively cleaned a record, I have found little need to re-clean. There are easy quick ways to do a "touch up" for any dust, lint, etc. that attaches to the once cleaned LP surface in handling and playing. Since many of the records I've bought are hard to find, expensive or both, they often need to be cleaned properly to play quietly.
There are countless ways to approach this--but I'd say there are good practices that are generally observed by archivists, which is a good starting point. Beyond that, some of the cost involved in the cleaning hardware is aimed at convenience and minimal effort. There's a happy medium that each person can choose based on budget, time, effort and results.

