As the achievement says, you need to own every property down one side of the board. Community Chest, Chance, the taxes, and the corner properties obviously don't count.
I got this quickly by playing against the AI on difficulty 1, and I made a trade to get properties from the AI. You don't have to win (or even finish) the game to get this to pop, so just purchase as many as you can and then trade for the rest. It helps to modify the default rules to have you start with as much money as possible and maybe even start with the properties divided between the players.