What is the easiest way to get to Turks & Caicos when coming from Manila, Philippines?

I'd like to go and visit Turks and Caicos but I don't know the easy access or procedure from my place in the Philippines

You would need a US transit visa if you stopped in Miami. You might want to go from Manila to London and then from there to the Turks and Caicos Islands, since it is a British territory and they have direct flights to the UK.

That's a long haul, and I hope you have a good reason to go there because there are so many more fascinating places that are much easier and less expensive to access from where you are.
Be that as it may, you must first reach Miami which is the hub for the Caribbean, and, then, there are many choices to reach your destination.
If you do have second thoughts consider the Maldives, the Seychelles or Mauritius as examples. Even closer to home, the islands and beaches of southern Thailand.

I've put two pages below. One on the T&C's and the other where there are flight tools and online flight booking agents. I'm sure you'll find a way.