This is a great scenario, I haven't liked it that much in my first game. I beat it yesterday as France.
First, I've built two settlers, then Ironworks and National Treasury in the capital. The French trait is centered towards City States, so I took Patronage and a Commerce or Freedom opener. The initial caravel scouted the shore of the New World and found a couple of Cultural tribes I befriended immediately. I landed the first settler near a first luxury I found, then a second one in similar circumstances, next to my first colony. By the time I beelined and built Naval Academy (removes Scurvy), I had two caravels reach China, for 150 VPs total (the first to reach China was England, and only 3 first caravels count).
By that time I met the Incans who were way ahead in score (as was England). I the beelined Musketeers and started churning out Musketmen, Cannons nad Conquistadors in the capital (2 turns on production focus, sometimes 1 turn with hammer spill). I then shipped the army to the New World.
Meanwhile I found Cerro de Potosi, which grants a Treasure unit every ~10-15 turns. Each generates 50VPs. I settled it immediately. From that beachhead I launched an invasion of the Inca land, taking one city and the capital. Razing the city created a Treasure which I had returned to the capital, but I stopped before the city was removed (so it would count towards my VPs). Capturing Cusco created 3 Treasures immediately. I returned 2 of them to the capital, when I took the last (third) Incan city and my points count was above 1000. This was a great game, full of flavour and excitement - Hiawatha and Monty, who were on another continent, were close to winning.
I think that conquering at least one Native civ is compulsory - it's a fast way to gain lots of victory points, fast.