A lot is going to depend on what DAW software you use to record and edit audio, which you did not mention in your post. You also did not mention how you intend to use the BC1 - to process input while recording, or to process already recorded material (or both). I would recommend that you re-read the Fireface manual dealing with signal routing and connections, pages 51 and 53, and with Totalmix, pages 51 - 77. There are also some informative videos online at YouTube dealing with making connections to and from the Fireface 800.
If you intend to use the BC1 to compress incoming audio prior to recording it, then you simply insert it between your mic pre-amp(s) and the Fireface. Of course, if you are using the Fireface's pre-amps, then you will need to use TotalMix to route the signal through to one or two of the Fireface outputs, which are then connected to the BC1 inputs, then routing the BC1 outputs to two Fireface inputs and thence to your DAW.
If you intend to use the compressor on material that is already recorded, then you need to route audio out of your DAW to one or two of the Fireface's outputs - use TRS to whatever the BC1 uses for connectors (XLR or TRS). Then route the output of the BC1 back into two inputs of the Fireface, and then use your DAW software to route the compressed audio to your main mix bus.
Making a round trip loop with this audio will introduce some processing latency (delay) to the audio you're processing, since it is passing through the Fireface D/A converter on the way out, and then through the A/D converter on the way back in. This is in addition to what processing delay the BC1 introduces, if any. You will need to determine this amount of time, and adjust your DAW's Latency Compensation so that it introduces this amount of compensation to everything that is not passing through the BC1, so that everything lines up.
In the Pyramix DAW, which I use, there is a handy little plug-in which is designed specifically for routing signal to and from an external piece of equipment. In it, you can determine the inputs and outputs of the Fireface to use, as well as what Latency Compensation needs to be applied. Surely other DAWs have something similar...
If you intend to use the external side chain input of the BC1, you need a Y cable with TS connectors on the two ends of the "Y", which get combined to a stereo TRS connector on the other end, like the second one shown in the manual on page 97. This will allow you to send stereo material from two outputs of the Fireface to the BC1's external side chain input.
Frank Lockwood
https://LockwoodARS.com
Fireface 800, Firmware 2.77
Drivers: Win10, 3.125; Mac, 3.36