Modeling and observation both were used. Basically galaxies were observed and there recessional velocities were measured (not at all easy to do) and this was graphed on the y-axis of a graph and on the x-axis the distance of the galaxies was plotted. Then a linear regression was plotted and the slope is equal to Hubble's constant (Ho) and was used in the equation V=HoD (V=recessional velocity, D = distance) The value is around 72 or so (km/s)/Mpc I believe.
From here you take the equation V=D/t and solve for t
Vt=D
t=D/V
----and then take the equation V=HoD and rearrange the variables so that one side is D/V
1=(Ho)/V
1/Ho=D/V
----and then substitue the vaules to get the equation: t=1/Ho
Taking the value you get from this you multiply it by 1/(# of seconds in a year) and by 1/(number of kilometers in a megaparsec(mpc)) in order to remove the km and mpc units on the value and you are left with the age of the universe.