Get a mono adapter
for your headphones and try plugging them into your Tech 21 Blonde. I haven't tried this with the Character pedals, but I think they have enough power for low-z phones.
Place any buffered effects pedals, such as digital delay or reverb, after the Blonde and then plug the headphones (with the mono adapter) into the last pedal in the chain. TB dirt pedals can go in front of the Blonde. I do this with a couple of Boss pedals and a SansAmp DI with great results.
Make sure your headphones are low impedance and high sensitivity. Your Sonys should work fine. Play around with the gain and volume on the Blonde for max output. If your headphones aren't loud enough, you can find louder ones for $30 or so.