How do you check how many cones you have in your eye?
One of the two confirmed cases of tetrachromacy was confirmed genetically:

From this Popular Science article:


  Based on Antico's genes, Jameson has determined that Antico's fourth
  cone absorbs wavelengths that are "reddish-orangey-yellow, but what it
  appears to Concetta is uncertain at the moment," she added. Since the
  tests aren't calibrated for this wavelength, empirically demonstrating
  tetrachromacy is still really difficult.


Color vision is connected to the X chromosome and mutations in the X chromosome cause less or more color vision.  In the above case, the theory that people with 2 mutated X chromosomes could have four cones instead of three was tested.

This image from Scitable by Nature Education articles simulates regular vision on the left and tetrachromatic vision on the right:



As far as how to test to see how many cones you have, there are some online test that may or may not be very reliable.  This Metro news article has a test produced by Professor Diana Derval, author of "DesigningLuxuryBrands: The Science of Pleasing Customers’ Senses.  However, due to the limitations of computer screens it would not be reliable to test for tetrachromacy.