What can I do in Hong Kong with a six-hour layover?

It's going to be quite short, but should be doable. Whether it's sensible is another matter.

Consider the following, though:

• your incoming flight may of course be late
• time to disembark (if you're in the last row of an A380, that could take a while)
• time to get to immigration checks (may be 2 minutes or 15 depending on your arrival gate, possibly more — from some gates you'll have to get down to the shuttle train below the airport and back up). Don't forget to fill in the form.
• queue for immigration (very variable, and even though I've never seen anything like the huge waits you can expect in some US airports, it can take a good 15 minutes+ if you arrive after a couple of other large aircraft, which is often the case around this time)
• I'll consider you have your baggage checked through
• withdraw cash
• buy Airport Express round-trip ticket, or better yet, an Octopus card (the reloadable contactless card that you can use to pay for the Airport Express, the MTR and buses if you use them, the trams, the ferry, and even your purchases at 7-Eleven, Circle-K and a few other outlets). You'll need one card per person.
• get to Airport Express (just across the arrivals hall)
• wait for Airport Express (every 12 minutes in general)
• Airport Express ride to Hong Kong (25 minutes)
• you're there!

You get right in Central, underneath IFC Mall. Great shopping and dining, though possibly not the most original part of Hong Kong, and most of it will probably not yet be open at that time. You can still eat dim sum at Tim Ho Wan, a small casual restaurant near the MRT entrance (the original Tim Ho Wan is known to be the cheapest Michelin-starred in the world, by a mile). Check out the view of the bay from the IFC Mall terrace.

Exit IFC Mall, use the elevated walkways so typical of Hong Kong. From there, you have multiple choices, but one that would probably get you a good glimpse is to take one of the trams (you need to pay with your Octopus), possibly towards Causeway Bay. Early in the morning might be relatively quiet in terms of things to see, though the market next to Times Square (around the corner of Bowrington Road and Wan Chai Road) should be an interesting sight (and smell). The Discover Hong Kong site (official tourism board site) recommends the Chun Yeung street market which is a little bit further.

Another option, but that's probably really pressing on time, is to go the ferry pier instead, and hop across the bay to Kowloon / Tsim Sha Tsui. You can pay for the ferry using the Octopus card. Note that the "Kowloon" station of the Airport Express is quite a distance from there.

For the trip back, you just do it in reverse: Hong Kong station, Airport Express, go through immigration (don't forget to fill the details on the copy of the form that is in your passport), security, get to your gate.

I think all in all, you should count about 1h30 on arrival, and about as much on departure, though depending on your airline's boarding policies, class of travel, frequent flyer status, and experience navigating around large airports, cities and public transport systems, you may need to budget for more time.

If you are unhappy with the time factor of getting into Hong Kong city, consider as an alternative: the Ngong Ping Cable Car to the Big Buddha. The bottom of the cable car is a five minute taxi ride from the airport (you'll need a blue taxi). The cable car ride is 25 minutes, and the Big Buddha statue at the top well worth seeing.

Allow a bit longer for getting back tot he airport than you took getting there, as blue taxis are sometimes hard to find away from the airport. Also there may be a wait for the cable car, unless you are prepared to pay premium prices for 'priority boarding'.

I have few things to add to Jcaron's answer. Returning your octopus card will cost you money. I'd advise on buying same day return airport express ticket. Second, exchange for some Hong Kong dollars and ask the MTR staff to change some coins for you, they're willing to do this in general. Third, you probably will hit peak hour traffic on your way, and you probably will see people rushing to work, if it's a weekday.

This is comments about the other answer:

1. Same day return ticket of Airport Express and Octopus fare are priced same.
2. 7am-8:30am is busy hour of MTR, the cabin looks like canned sardines.
3. Tim Ho Wan opened at 9am, you would be better to visit other places first.
4. Many of the local market opened at 10am or later, except those selling food. So it is nonsense to visit Temple street/Women street.
5. Ngong Ping Cable Car opens at 10am on Weekday and 9am on Weekend/Sunday, but you may go there by Bus, they start business at 7:15am on Weekday (1hrs later on Sunday)

Tripadvisor listed out all the thing you can do in HK. There are some places not mentioned before but suits your time.

1. Nan Lian Garden & Chi Lin Nunnery
2. The Peak and the Peak Tram
3. Wong Tai Sin Temple
4. Flower Garden Street (9am-later)

And finally some transport website may help you

1. KMB Bus planner
2. MTR planner

Keep in mind that you will have to report the purpose of your visit to Hong Kong when you go through immigration, and the immigration officer may not let you through if you only have a 6-hour layover. I once had a 4-hour layover in London Heathrow, and I wanted to leave the airport to see a little bit of the city, but the immigration officers wouldn't let me leave. Maybe 6 hours is long enough that you won't run into this problem.

Unfortunately, I'm not sure if there's any way to know in advance whether you will be allowed to leave the airport.