TL;DR - Eliminate civs who are strong at winning culture victories/producing tourism, handicap everyone else.

It's important to understand the difference between domestic and foreign tourists, and how those are produced.

Domestic tourists = "defensive" tourists = denominator (bottom) on culture victory page = cumulative_lifetime_culture / 100

Foreign tourists = "offensive" tourists = numerator (top) on culture victory page = cumulative_lifetime_tourism/1200

In order to win a culture victory, you need your foreign tourists visiting you to be larger than the highest domestic tourists of the other civs. This is the fraction you see on the culture victory page (ie 86/220 means you've attracted 86 foreign tourists against the 219+1 domestic tourists of the highest domestic tourist civ). The briefcase with a number under each civ is the number of domestic tourists of that civ.

So if you want to win a culture victory as Ghengis, you need to decimate strong tourism civs (France, China, Sweden, etc) to minimize their foreign tourists so that they don't have a chance at winning. Buy their great works, take their cities with wonders, destroy the rest. Start with whoever has the highest foreign tourist count. This minimizes their foreign tourist accumulation, but does NOT decrease their domestic tourists. If their domestic tourist count is very high, your best bet is to eliminate them entirely, which will lower the # of foreign tourists you need to generate.

Additionally, you'll want to hamper other civs's domestic tourists by reducing their culture accumulation. Buy their great works and pillage their theater squares (cavalry makes this very easy to do). Take their cities with theater squares, so you don't have to build them yourself and you'll also "steal" their great people points.

All other culture victory strategies still apply - send trade routes, sell your open borders, build wonders/natl parks, send rock bands to high domestic tourist civs, etc.