Dell has several methods for updating their BIOS.
Out of all the different approaches, this is what I cam up with that successfully let me install the latest BIOS without taking a trip to the data center.
./PER610_BIOS_LX_2.1.15.BIN --extract bios_files
would extract the files to a directory named bios_files
dellBiosUpdate -u R610-020115C.hdr
A GUI that's a part of the Lifecycle Controller. I think you get to it by pressing F10 for System Services during boot.
Dell's Drivers and Downloads page has BIN files you can download and run from within linux to update the BIOS. These are a combined shell script and archive. In theory, you just run them and the new BIOS is installed.
Dell officially maintains and supports a yum repository with rpms for their firmware and some tools for installing it.
Matt Domsch from Dell maintains an unsupported yum repository just with BIOS updates. You need to install some other tools from Dell's community repository (or maybe smbios-utils and firmware-addon-dell from EPEL would be enough, haven't tested this yet) to use it.
Update: vwaelchli on reddit points out that you can also update the firmware by running wsman commands from the lifecycle controller.