Last time I was in London I spent about 1.5 hours standing in line in customs. That could cause you a problem. Can you more your flight back a few hours and get more time?
Anyway, Heathrow Express is the quickest way to get into town. They leave about every 15 minutes and take about 15-20 minutes to get to Paddington. The problem is when you get to Paddington there's not a whole lot to see there. It's about a 10 minute walk to Hyde Park but that's it. So, you'll need to get on the tube. It's pretty easy to use. Most people use something called an Oyster Card to pay for the tube but since you're there such a short time, you can buy individual tickets at kiosks in the stations. Of course, you need to decide what you want to do. You could go to the British Museum, spend the whole time there, see a lot of cool things but not really see any sights, The Museum is free and one of the best in the world. I'd guess you'd want to see more of the city, so I'm throwing these out:
For all, take the tube to Westminster.
[ol]
Wonder around Parliament Square, Visit Westminster Abbey, Visit a Pub for a beer and bangers and mash and Churchill war rooms.From Westminster tube, walk to the river and get on one of the cruise boats that go up and down the Thames to Greenwich. You'll see a bunch of sites and I think the boats are better than the hop-on hop-off buses. The buses can get caught in traffic and be slow.Wonder Parliament Square, walk up Whitehall to Trafalgar Square, walk down the Mall to Buckingham Palace, walk behind the Palace into Hyde Park.[/ol]
Option 1, you get to see two of London's best attractions but it's a little pricy for each one and you will not see much else.
Option 2, you get to see a bunch of stuff but you're sitting on the boat the whole time.
Option 3, it's the cheapest and the walking is easy. You'll through the coolest parts of London.
I only mentioned a Pub in option one but I'd go to one just to soak up some London culture. They're everywhere and you won't find a bad one in that area.
Pay attention to your tube map to get back to Paddington.