To determine the age of the universe one needs to know:
1) the Hubble constant, H_0, which is the current rate of expansion
2) the current densities of the different matter and energy components of the universe (density of baryons, cold dark matter, radiation, dark energy)
3) a model for evolving back in time
The Hubble Space Telescope provides one of the most accurate determinations of H_0 today; measurements of the cosmic microwave background (CMB) by a variety of instruments, including NASA's WMAP satellite, have furnished reliable determinations of the densities of the various matter and energy components of the universe. Einstein's General Relativity, under the assumption of spatial homogeneity and isotropy gives the Friedmann Equations -- the theoretical model for the evolution of the universe. These are used to determine the universe's age.
The reference frame used in these calculations is the rest frame of the CMB.