You can view the BeagleBone Black boot process, and login via a serial connection (with a TTL adapter) by connecting to the serial debug header. The pinout is as follows:
J1 = Ground
J2 = Not connected
J3 = Not connected
J4 = TXD
J5 = RXD
J6 = Not connected
Your serial port should be set for 8N1, with hardware flow control disabled. The baud rate defaults to 115200.
I'm using a PL2303HX based USB to TTL converter. Units like this can be purchased very inexpensively.