I think the plot you show is the estimated abundance of the interstellar medium from which the Sun has formed.
The chemical abundances of the interstellar medium change with time, so you have to define some point in time at which to estimate them. As the initial chemical abundance in the Universe is basically H, He, with traces of D, Li and Be, then it clearly isn't the primordial abundances as all heavier elements are made in stars.
So how do we know the abundances of the ISM from which the Sun formed? It's a different story for each element, but basically they fall into two categories: either the abundances come from a detailed analysis of the spectrum of the Sun; or more directly by analysing the composition of meteorites. Full reviews of this can be found in Lodders (2003), an Asplund et al. (2009).
Many elemental abundances can be measured with extraordinary accuracy in meteorites that have fallen to Earth (usually with respect to Silicon), having formed from the protostellar nebula. This is usually taken as the best method, but there are many chemical elements that are not properly represented in meteories - for instance volatile elements H, C, N, O and the noble gases. For these elements the solar spectrum can yield less precise abundances from a detailed knowledge of atomic line transition strengths and the structure of the solar photosphere. In the case of Li, it is the other way around. The Sun is Li-depleted and the abundance comes from meteorites. The solar and meteorite abundances are put on the same scale by matching the Si abundance (and sometimes additional elements).
It is interesting that your plot has no error bars. Some of the elemental abundances are afflicted by significant uncertainties. Most meteoritic abundances are known to 5%. Some of the solar abundances are uncertain to 10-50%. There is usually reasonable agreement where there is no reason to expect that one or other of the abundances has been altered by some physical process. Of some concern are the C,N,O abundances - these are among the most abundant elements in the present-day cosmos (O is 3rd behind H and He). Yet the abundances of these are known (from the Sun) only with a precision of 12% (1-sigma).
Edit: Of course the abundances found from the Sun and meteorites are not necessarily the abundances of the present day interstellar medium or the abundances somewhere else in the Galaxy (or indeed other galaxies). But similar techniques can be applied to stars of different ages or in different places or even to excited gases in the interstellar medium. Of course what can't be done is to actually get hold of some of the material and analyse it, so generally speaking, abundances determined for "elsewhere" are less precise than for the solar system or Sun.