Doing a criminal check can be tricky.  Here are a few resources to get you started:

Zimmerman’s Research Guide: Criminal Background Checks (from LexisNexis)
Provides a quick overview of conducting criminal background checks–what you will and won’t find, as well as sources.

BRB Publications: Free Public Record Sites
This is a listing of free government public records sites, by state.  For NY, see NY State Inmate Information (includes former and current inmates).

The Virtual Chase: Art of Public Records Research
A nice guide to doing public records research with some pointers to sources.

Note that criminal records are going to be state-by-state, and what’s available will vary by state.  The best way to go about this is to figure  out the states in which the person lived and then look to see which databases are available for that state.

Federal and State Cases via LexisNexis
Search by case name, citation, or keywords. Cases include annotations and links to related cases.
Guide to finding cases in LexisNexis

FindLaw.org
There’s a wealth of information on this site including links to information about the Supreme Court, court cases, and codes for the different states.
To find Supreme Court Cases, go to http://www.findlaw.com/casecode/supreme.html.
For other helpful legal resources, go here: http://lp.findlaw.com/

NYU Law Library Legal Research Guides
NYU Law Library has a number of helpful research guides including A Guide to the U.S. Federal Legal System: Web-based Publicly Accessible Resources that includes links to finding Supreme Court Cases.

In addition, the 6th Floor Reference Desk at Bobst which specializes in Business, Governement Documents, and Public Policy can be a good starting place for law-related matters.