I'm no expert on copyright but I can give you some of the basics in searching for renewals using the volumes on archive.org. Works published in the U.S. between 1923 and 1963 were granted copyright protection for 28 years. To extend that protection copyright would have to be renewed in the 28th year. For works published after 1963 the renewal was automatic. So, the records that concern us are from around 1950 to 1992. Renewals filed after 1977 can be found by searching the U.S. Copyright Office's online catalog. Those filed prior to 1978 can be found in the volumes of the Catalog of Copyright Entries (CCE) which we can access on archive.org.
CCE volumes were published every six months. From 1947 through 1956 renewal registrations were published separate from current registrations. Although each volume covers six months the files on archive.org have been combined into complete years. Each volume is alphabetized by the work's title. There is no name index. You can use archive.org's online reader to search for keywords but this isn't a foolproof method as OCR and the scanned images aren't perfect. If you're searching for "Brian" you might also want to try searching for "Br1an."
Starting with the 1957 volume renewal registrations were published in the same volume as current registrations but within a separate and of course much shorter section. These volumes also include a name index that combines both renewals and current registrations that cross reference the titles. Due to the length/file size of these volumes they have not been combined into complete years so both 6 month volumes have to be searched separately.
Beginning With the 1974 volumes each six month volume was split in two and included a name/title index at the beginning of the first book that cross reference the work's registration number. The listings of current and renewal records are by registration number.
It's best to search the volumes for the year prior to the 28th year, the 28th year, and the year after the 28th year (renewals filed late in the 28th year will show up in the volume for the following year). Additionally, you will want to check to see if a Notice of Intent to Enforce (NIE) was filed for a particular work. You can do this through the Copyright Office's online catalog http://www.copyright.gov/records/cohd.html
and also check the files here http://www.copyright.gov/gatt.html
I hope this was the type of information you were wanting.