I wouldn't want to be dumping bleach everywhere...
I had out of control ivy on my chimney and vinyl siding. I found that Round Up would slow it down a bit, kill a few leaves, but leave the plant intact and still growing.
What I had to do was cut the ivy at the ground--it was as thick as a small tree trunk in a few places, then I dug down and pulled all the roots out using straps, come-alongs and my car as an anchor point.
After I did that the parts that were on the house/chimney died within days and I was able to pull it down.
A word of caution--
You will never be able to get the "suckers" out of the wood, vinyl, chimney... it's been 2 years and I'm still looking at them even after spending days that first summer scrubbing... It would take me about 10-15 mins to clear a 1'x1' patch of vinyl... the brick required a wire wheel brush and after hours I just gave up. Now I wish I had left it up but done a better job of containing it... in fact I often consider replanting some to send it back up the chimney.
My advice is based on anecdotal knowledge or personal experience. I'm not a professional no matter how matter of factly I may say something