How to get free programmatic access to citations counts for a given paper?
If you are an R user, you can use the scholar package. This package allows you to analyze data from Google Scholar and obtain citations, publication list and even perform predictions of the h-index. An example R code:

library(scholar)
id = "xJaxiEEAAAAJ" # Isaac Newton's id
cit=get_citation_history(id)
barplot(cit[,2],names.arg = cit[,1])