What is the best way to bid on a website domain auction?

Not enough information. Tactics depend on various factors:

(1) the venue, since auction formats differ greatly across domain market places;

(2) which of various auction types at a given website you're looking at;

(3) listing settings decided by the seller such as "buy now" options or a minimum reserve;

(4) the seller's past habits with regard to promoting, altering, or even -- unfortunately -- rigging auctions;

(5) Quality of the domain itself relative to the price.

Without knowing the details, no general answer that applies to all situations can be given.

I acquired Referable and other big domains by reaching out to the owner directly when they were not listing it. That's the best way to get a discount. Regarding Flippa, if you bid and it doesn't get accepted, try following up with them in 2 months via LinkedIn, email, or phone (search Google for WHOIS DomainName to find contact info). They may realize by that point that they are not going to get as much money as they were hoping for and be willing to part with it for less - especially if you establish a personal relationship and rapport, and they believe in your vision for the URL. Funny, but it matters to many people (domain squatters not included).

If this is something that will improve your business, allow you to start a business, or in some way will have a positive impact on what you want to accomplish, then trying to game the auction won't be effective. First and foremost set what you are willing to pay for it in your mind based on what value it will have for you short and long term.

If it is a revenue producer, then if nothing happened to improve it, how long will it take to get your money back.

If it has some other positive impact, then determine what that value is to you and the very highest possible level and then set a dead hard last price on it.

On Flippa, I typically wait to see where things are at the last few hours. If it's got value to me and the bids are in my range, then I jump in at the last hour and see where it goes from there. Flippa extends the auction automatically though, so there's no value in sniping.

The best way to bid on a website domain auction or any online auction is to watch the deadline closely and bid more than the competition.
Best of Luck,
Michael T. Irvin
michaelirvin.net
My books are available exclusively through Amazon Books. Check out my book "Copywriting Blackbook of Secrets"
Copywriting, Startups, Internet Entrepreneur, Online Marketing, Making Money