You shouldn't need to boot from the floppy drive. You should just put the downloaded bios onto a floppy and enter bios when you restart your computer, usually the DEL key. Within bios find the option to upgrade your bios image and point to your floppy for the bios image file.
This is nice on new boards, but not on all boards and for sure its not on older boards.
Once in DOS.
Type a: <Enter>
at the a:\> prompt enter the following without the quotes
"awdflash m615b_15.bin" <Enter>
This should start the flash.
I personally use a Windows 98 CD as a bootdisk for this type of thing. then the floppy for the flasher and rom/bin file
UBCD also has freedos that should work, but not ALL software is compatible so use at your own risk.http://www.ultimatebootcd.com/
Here are some DOS boot CD images. I have not tried them. Also note, DOS can not see NTFS drives. So you will most likely only see the a: (flpppy)http://www.bootdisks.us/ms-dos/5/ms-dos-bootable-cd-images.html