This achievement requires both quickness and a bit of good luck.
1) Skip the tutorial and run right collecting all the coins.
2) Build the camp. Recruit the two beggars. Buy 2 bows.
3) Run left until your reach the camp. Recruit 2 beggars.
4) Return the the camp. Collect coins from your archers.
4) Run to the right until you reach 2 more camps and recruit 4 more beggars.
5) Return to the camp. Collect all the coins. Buy 6 bows.
Hopefully, by the time you return to the camp for the last time, you will have enough coins to buy all the bows.
If you are luckily enough, there may even be a treasure chest between the two camps on the right, which makes this task a bit easier.