This article reasonable accurately gives the highlights of the history: http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/measure_c.html
As stated, there is no uncertainty of the definition because the standard was reversed, the meter was redefined in terms of the speed of light in a vacuum as opposed to the speed of light begin defined relative to a meter. c, the constant is the standard so is by definition exact. If c is more accurately measured in the future, the accepted length of a meter will be what changes, not he number used for c.
As to how c is measured, there are a number of ways, some mentioned, but the accepted number was basically reached from extrapolating different methods and reaching a common answer. For instance, air has a know, measured index of refraction which effects the speed of light. If the speed is measured at one standard atmosphere, then again at 0.5 atmospheres, at 0.1 atmospheres, at 0.01 atmospheres, etc., and the graph is found to be accurately linear, then one can extrapolate to 0 atmospheres. If this technique is used with high accuracy measurements for multiple different methods of measuring the speed of light in different media, different wave lengths, etc. and they all agree to measurable limits, then the number is assumed correct to within measurable limits.