I'm already backing my entire computer (Mac) up to an external drive using Time Machine but I have a fireproof hard drive that I put my most critical documents onto, as well as my photo library. The way I used to do it with Aperture was use the vault functionality which just backed up the changes to my library when ever I told it to. However, I can't seem to find a similar option in Lightroom which means that every time I backup I have to just delete the photo library off the hard drive and recopy the entire thing. This takes about 2 hours rather than the 2 minutes Aperture would take. Also, I can't use Time Machine because this drive doesn't have nearly enough space to backup my entire Mac, just the important stuff.
Really this has nothing to do with Lightroom or Photography. The solution is the same regardless of which type of files you need to back incrementally: rsync
This a utility to synchronize file-systems by only copy differences. It can do quite a lot actually as there are dozens of options. I use it on Linux as part of my backup procedure. A simple command such as:
rsync -Sax /pictures /backup/pictures
Does the right thing most of the time for me. Check the documentation online from Apple to see what is the right command from you. In my workflow for images, nothing ever gets deleted or overritten. For cases where you delete files, you can add the
--delete flag too.
If you don't wanna use TM to back up to the fireproof drive, then consider other backup options. If you don't wanna use a command line tool like rsync, consider other backup software: there's Data Backup by Prosoft, Mac Backup Guru, and maybe even one of the cloning applications like SuperDuper or CarbonCopyCloner. They can make incremental backups or clones, probably using rsync under the hood. Or even just drag and drop; IIRC the Finder will ask whether you want to replace all or just add. BTW, Data Backup can do versioned backups, as TM does, as well as clones and incremental copies.
It isn't clear from your query whether it is the Lightroom catalog or the photographs themselves that is the issue in regards to backups.
All your photographs should be under a single directory and can therefore be incrementally backed up using Time Machine, File History (for Windows) or using rsync as mentioned in a previous reply. This is setup independently of Lightroom - remember Lightroom does not store any of your photographs it just indexes them (and may import/copy them to your photos directory).
The Lightroom Catalog is essentially a single database file (there are some other cache files too but they don't need to be backed up). A backup of it is essentially a copy of the whole file - there is no incremental option and neither is it necessary since it isn't a very large file. You can specify catalog backup options in Lightroom and have those backups be on a different disk to your main working disk. I set mine to produce one backup per day (when I am using Lightroom).
If you are looking at offsite options for you photographs that would be a whole bigger question. Personally I don't consider any of the cloud options suitable or cheap enough yet for amateur use so backup to a portable HD which I store at work. If I was a professional then I would use one of the professional paid for cloud services.