How can I get a list of citations for a DoI or PubMed ID?

Quick and dirty version: Google Scholar

Type-in the DOI and read the citation count on the field "Cited by". Google Scholar does not filter for quality, so the citation count will include self-published papers, fake publications, etc. Expect some noise.

Better version: Web of Science

Ask your institution's librarian how you can access the Web of Science database and how to use it to retrieve the data that you need. That might require you to walk in daylight to the physical location of your library. Many survived that journey.

The graph you are looking for is not easy to get and therefore probably not publicly available for free. Certain firms have specialized in making this data available (e.g. Thomson Reuters "Web of Science, Google Scholar). To protect their asset it seems likely that they will not let you download their databases or query it in such large quantities.

In a way, their business model depends on making this hard to access, for if you could download the whole database, you could offer the same service for a lower price.

Maybe even their terms of use forbid too unstructured or too large queries.

As for doing this programmatically, there are two options. Daniele Rotolo has written an R-script (MedlineR) that will allow to download Web of Science (WoS) information from PMIDs, if your institution provides IP-access to WoS. See: http://www.ncbi.nlm.nih.gov/pubmed/15284107 However, there are some issues with this program - it occassionally crashes after fetching ≈1,000 records.

The other (better) option requires your institution to subscribe to Thomson Reuters Web Services Expanded, which will allow you to use a SOAP API to search WoS and download relevant fields. This also includes cited reference searches. If you can confirm that your institution has such an access, I will be glad to share my own (simple) PHP SOAP call for matching PMID against WoS accession numbers (UT).