The problem of evil doesn't presuppose that God exists. I'll explain why.
Problem of evil is a proof by contradiction.
Proofs by contradiction go like the following:
>Let's hypothesize {X}
>if {X} then {Y}
>but {Y} contradicts {X}
>so {X} can't be true
For example:
Let's hypothesize that 1+1= 1
1) 1+1 =1
2) 1 = 1+1
Adding (1) and (2)
3)1+1+1 = 1+1+1
Dividing both sides by 3
4) 1 = 1
If 1+1=1 then 1=1
But 1=1 contradicts 1+1 =1
So 1+1=1 can't be true
Now for the problem of evil:
Let's hypothesize that a God(who follows his own morality) exists
1) we assume that God's morality is the same as the Christian morality for convenience
2)by christian morality it is immoral to allow evil (by christian standards) to exist
3)there are instances of evil that God has allowed to exist
4) God is immoral by his own (christian) standards.
If {a god who follows his own (christian) morality exists} then {God is immoral by his own (christian) standards}
But {a god who follows his own (christian) morality exists} contradicts {God is immoral by his own (christian) standards}
So {a god who follows his own (christian) morality exists} is false
>God is by definition good
God has no motive to do anything besides his plan, and since his plan is the way things ought to be, then it is good. But that implies that we are incapable of evil, since it's all part of God's plan anyway. If a serial killer goes out murdering, God allowed it, so it is good.