The category of flat vector bundles is equivalent to the category of local systems, see for instance [this MO-question,][1] which in turn are equivalent to representations of the fundamental group, see [this MO-question][2]. Via these correspondences, the vector bundle associated to the covering corresponds to the representation of the fundamental group of $X$ on the fibre of $f:Y\to X$ via deck transformations. In particular, the vector bundles associated to coverings are flat. 
A flat vector bundle has trivial characteristic classes, but triviality of characteristic classes is not sufficient for flatness, as the answer of abx shows. 

  [1]: http://mathoverflow.net/questions/4138/
  [2]: http://mathoverflow.net/questions/17786/