The problem that I have with this "classic argument" is that two of the assertions made are probably false, which invalidates the argument.
The first error isn't even in the list of five. "Christians believe in five things." Actually, they believe in a lot more than those five, which are conveniently ignored for the purpose of this exercise, but some of those get you into trouble.
For example, consider #1. "If the Christian God exists, he knows all about evil and has the power to stop all evil." This is not necessarily true by God's choice, given that the Great Test is whether or not, given free will, an individual chooses Good or Evil. Free Will is a limitation that God has placed on Himself. So God does *not* have the power to stop all evil. He relinquished that when free will was granted to humanity.
God has also promised never to flood the earth again, or so I'm told (something about rainbows... *grin*), so clearly there are things which God chooses not to do.
The more obvious error, however, is #3. "If the Christian God is all good, he should stop all evil and it should not exist." Whoa, Nelly! This assumes a lot and presupposes you understand the Mind of God, which most Christians would assert you do not. By #2 (God is all good, which is "by definition"), anything God does is good, even if it is the exact opposite of what He commands us to do (i.e. do as I say, not as I do). Thus, anything which God is doing right now, even if it's not stopping men from committing evil acts against each other, is by definition good. It's good because God is doing it, and God is "all good".
I don't think you'll find any argument with #2 and #4 from Christians -- "God is all good" and "evil exists" are pretty basic components, I think. (Free free to jump in and correct me, Christians.)
Since we have two components of this four-premise syllogism invalidated, we really should throw away the remainder of argument and start fresh.
Be advised, however, that you won't get anywhere with attempting to "prove", logically, the existence or absence of (the Christian) God. Christianity is about faith, not logic.