how do i start an affiliate website without a big budget?

The money side is cheap: a domain, basic shared hosting and a free theme. The real cost is time. Pick a niche narrow enough that you can be one of the better sites in it within a year, then write the content buyers read right before they buy: comparisons, "best x for y", honest reviews. If you go with Amazon Associates, you need three qualifying sales within 180 days or they close the account, so don't apply until you have some traffic. And put an affiliate disclosure on every page with links.