The black streaks are grease which has been heated to the point of welding itself to the copper.
You might want to start with something more gentle, but when you need some cleaning thuggery spray the pan with Easy-Off or a similar, caustic oven cleaner, let it sit for a while, then wash it off.
Some oven cleaners only work well when heated, if that's what you're using you'll have to put the pan in the oven for a little while. Ahh the sweet aroma of the fumes...
Note that you need to remove the oven cleaner completely before going to any sort of acidic cleaner.
The method isn't exactly "green," but it's beaucoup effective and requires far less scrubbing than BKF or the home made chemistry methods.
Speaking of which... Flour is a very gentle and very fine polishing agent. It's great for removing tarnish and leaving a glow behind, but it's not terribly effective against the tough stuff. Ditto, the salt/lemon method and its variants; not exactly strong, but a BIT more aggressive than flour.
BKF is good stuff, but still not what you'd call really strong, at least not without a lot of elbow grease. You should absolutely have some around the house for all sorts of things, including cleaning the stainless steel surfaces it's designed to clean. Worth a try before going to the poisons.
Hope this helps,