How can I connect a BeagleBone Black using I2C to my TI host-controlled battery charger?
First thing you need to do is to connect the SDA and SCL pins of the charger to the SDA and SCL at the beaglebone. 
The beaglebone has 3 I2C ports. SDA0 and SDA0 is internal and cannot be accessed. SDA1 SCL1 and SDA2 SCL2 can be accessed. Do not use external pull up resistors as the beaglebone has inbuilt pull-ups.

To read the data on the beaglebone's terminal follow this:

https://www.youtube.com/watch?v=8C2zk6B-eLU

If you want to adjust a few parameters on your own then you could write a small C/java/python code with the following library.

https://learn.adafruit.com/setting-up-io-python-library-on-beaglebone-black/i2c

I can't help more since I don't know more technical specifications of your project. But this is the max I can do for now.