Some publishers provide altmetrics on their websites to authors and the public. Many large publishers are now collecting usage data or are partnering with other companies to provide this information to their users. PLOS has been a leader in providing altmetrics to both readers and authors. Each article they publish has a "Metrics" section which provides a wealth of information. Click the image below to see an example.
Many digital repositories such as Iowa State's Digital Repository @ Iowa State University (DR@ISU) provide authors with a dashboard (example below) that show statistics about their works in the repository. These statistics are separate from the statistics gathered by publishers* so if you are archiving your work in a digital repository make sure you take these into account! DR@ISU does display the Altmetrics badge for journal articles but it does not include downloads made through the repository.
Click the image above to see an example of an Author Dashboard.
The other place you can gather altmetrics is through online social interactions. Not all platforms are open, but Twitter for example, has an open API which lets other websites connect to it and harvest statistics. Retweets, likes/favorites, and shares are a way to track the interest level of a research project or article. Comments on blogs, reviews, articles, etc. are another place you can look.
Another source of altmetrics are social bookmarking and reference management tools. There's been some research that shows that the number of times an article is "saved" by users may correlate to later citations. (see Altmetrics in the Wild: Using Social Media to Explore Scholarly Impact for more information).