Tim,DNI is a document like an internal passport. People are applying for DNI after getting residency (radicacion). For radicacion you will need an apostilled original birth certificate. It can be a duplicate, issued by the same agency that normally gives them out, but it can not be a copy (not even a notarized copy). Since immigration takes your original birth certificate away in exchange for residency papers, and usually it gets lost, it makes sense to request a duplicate in your contry of origin.You will also need a document confirming that you don't have any criminal history in your country. Sometimes it is called criminal history report, sometimes it is called certificate of good conduct. Normally you can get it at a local police station. Get it apostilled as well.
Depending on the way you decide to pursue your residency, you may need some other documents as well I guess.