How to assign the signals in the H-NMR of Aspirin?
Here is a link to the actual assigned proton nmr spectrum for acetylsalicylic acid.  Your peak assignments are correct.


  Would E and B be doublet of doublets because they couple with D and C
  so they would have ortho and meta coupling?


Yes, that is correct. Here is another link with more detail on the coupling constants (see p. 13 for example)


  Would D and C be coupling with each other and B and E so would they be
  doublet of doublets of doublets, with two ortho and one meta coupling?


Conceptually you are correct, however in the case where two coupling constants are very similar, the expected quartet collapses to a triplet. Since $\ce{J_{DC}}$ is very close in magnitude to $\ce{J_{BC}}$ the expected quartet appears as a triplet.  Therefore, proton C appears as a triplet from these two protons and then each of these lines is further split into a tight doublet by the smaller coupling to proton E.  The overall pattern appears as a triplet of doublets.  The same reasoning applies to proton D, and again a triplet of doublets results.