I gained this achievement last night soon after joining gaming.stackexchange :-)
I focused on building improvements that made the biggest impact on culture (monument, temple, Mughal fort, etc.) and built as many wonders as possible (especially those that provide cultural boosts and affect the acquisition of social policies (Cristo Redentor, Hermitage).
I allied with as many Cultured city-states as possible (earning enough gold can be challenging) and picked the following policy trees (roughly in this order):
As others have said, gold is a problem. With all the wonders that I'd built and the bonuses you get for playing as Ghandi (less unhappiness for larger populations) Golden Ages were more easier to start. I boosted the duration of Golden Ages by building the Chichen Itza and Taj Mahal wonders and applying the Organized Religion and Reformation (Piety) social policies. Sometimes, I also used Great Artists and Great Generals to increase their duration. The additional gold helped a lot with winning over city states, buying buildings and negotiating with other civs.
One point that may not be obvious from the achievement's requirements: you can have more than 3 cities during your play through (though you may suffer penalties in acquiring social policies) - just make sure you raze or give those cities to other players before you complete the Utopia project.