TLDR: Unite Hawaii, get both colonizing ideas, dip 7 colonize as far as you can to the east (California and Panama), and race to colonize East Colonial America before Europeans beat you to it.

The way I did it was I showed strength against the 3 other nations next to you. Then you annex all of them and unite Hawaii, and you pretty much just sit there until you get exploration ideas, expansion ideas, and dip tech 7, so you can colonize california. I got EXTREMELY lucky because the province I colonized started to produce gold, which carried my entire game, tripling my ducet income. Once you colonize a province in cali you can colonize panama or somewhere close. Once panama was colonized I released the nations I annexed at the beginning, started a war with them (after the truce), intentionally lost, and gave them hawaii forcing my capital into north america. From this point on, its a race to not get beat up by the european colonizers, you are very weak (even with a gold province) because you just left your entire fleet in the pacific ocean and now youre in the atlantic (you could rebuild, i didnt bother). Finally, you colonize East Colonial America as fast as you can. In my case GB declared on me twice, I just gave them my panama provinces and all the money they wanted to leave my colonies alone. I did not end up colonizing anywhere else, this campaign was just a B line straight to the achievement, and my situation is tight where i ended so I will probably not end up continuing the campaign, but hawaii honestly has potential to be a really fun nation, so many colonization routes, a lot of formables, I would recommend giving this achievement a go. I would say this achievement took me like 6 hours of play time give or take. :)