I tried polishing a green marble tile, I think it was called Taiwan green. The marble was shiny but it did not have the depth that most marbles have. Is there a technique I can use to get the depth and clarity?

Many green marbles will not have the depth of shine that other marbles have. The greens from Asia are some of these types. The surface will have a good shine but little clarity.
Unfortunately there is not much that can be done to give the stone any depth of clarity. The overall shine can be increased by applying a small amount of marble polish or the stone can be treated with the recrystallization process. If you do decide to recrystallize, be sure the stone is throughly dry.
CAUTION Green marbles are very susceptible to spalling. Be careful using recrystallization since this process may block the breathing of the stone and promote spalling.