Will running my TB drive through my TB monitor rather than thru my Mac Book Pro port slow performance

If I'm running the display and the audio through the TB wire to my monitor as well, will that degrade the read/write speeds I can get to my TB drive? Will I get better performance by running the drive directly through the mac book TB port?

If your drive has TB though, put the monitor at the end of the chain. Your drive is not getting anywhere near the max throughput of a TB connection.

Grant,
What Ben says is a good practice except the actual performance is more complicated.
Which MBP you have also changes performance throughput. There's TB1 and TB2 and different chipsets for various models of computers.
For the rMBP, the 2 ports each have 2 lanes, so if you are driving a 4/5K display and running a multi-bay RAID, then you will be better off running each device off separate ports. If you daisy chain the devices, then it is total data bandwidth/port limited.
If you are just running a single hdd/ssd then there is plenty of bandwidth on TB1 or 2 and you will not see any improvements.
Anandtech has done some in depth test(s) of throughput on MBP TB ports.
www.anandtech.com/tag/mac

Hope this helps, Greg