You actually get the achievement, but the game then takes it away a moment later. To keep it, as soon as you get the achievement hit shift+tab to go to the Steam overlay. Then bring up the task manager and 'end task' the demo. Now you can't play the demo ever again, but it will show up in you game library with the achievement and count as a 100% completed game.
All weapons that can decapitate will work with the achievement, including those which do not buff you based on how many heads you've taken in a life. That means all these:
Eyelander
Scotsman's Skullcutter
Horseless Headless Horsemann's Headtaker
Claidheamh Mòr
Half-Zatoichi
It does make things easier if you equip weapons that help you run faster, like the Eyelander or the HHHH. I wouldn't use the Skullcutter for it.
Obviously, you have to be a Demoman to get the achievement using the Half-Zatoichi; it won't count if you're a Soldier.
First you have to own Flash; he's the robot Dog that costs 8,500 coins in the Gadgets menu. The achievement is only possible with Flash equipped to a Gadget slot.
All you have to do is travel 10,000 meters cumulative with Flash equipped. You don't have to reach 10,000 meters in one run, multiple runs still count toward the total. After you've gained 10km in travel with Flash you'll be awarded the achievement upon your death.
There's a counter in My Profile under Gadgets that lists "Total Distance run by dog"; this is the count that matters. The counter appears to go up whenever Flash runs, not Barry. I'm not sure if it goes up when Flash jumps yet.
Best Answer
You actually get the achievement, but the game then takes it away a moment later. To keep it, as soon as you get the achievement hit shift+tab to go to the Steam overlay. Then bring up the task manager and 'end task' the demo. Now you can't play the demo ever again, but it will show up in you game library with the achievement and count as a 100% completed game.