This is easiest on a Tiny sized Archipelago map. Play as Joao III, and select Pedro II, Hojo Tokimune, and Gandhi as your opponents. Once you meet all 3 fellow leaders, simply send a trade route to each Civ, and allow for enough time for each trade route to finish. Note that if one of the Civs does not have a coastal city, it may be wise to create a city near the landlocked civ and gift it to them, securing an easy location to trade with. Once the last trading post is created, the achievement should be unlocked.