Does The Bible Say God Created Evil (Sin)?
KJV: I form the light, and create darkness: I make peace, and create evil: I the LORD do all these [things].
This generates some confusion, because the "evil" God creates does not necessarily match our definition of "evil" (more on this in the next section). We can try reading other translations and commentaries on the verse to gain some insight. A good source for comparison and commentary of individual verses is biblehub.com. The Hebrew word ra` (Strong's H7451), there translated as "evil" in KJV is translated as "calamity" (in NKJV, KJV 2000, ESV, and NASB), "disaster" (in NIV, HCSB, and ISV), "woe" (in RSV) and "bad-times" (in NLT). It is commonly believed to be referring to punishment, and not unrighteousness (see Matthew Henry Commentary for Isaiah 45:5-10).
An example of a similar usage can be found in Genesis 19:19, where ra is again translated as "evil" in KJV but in more modern translations (including ESV, NASB, NIV, and others) the word is often translated as "disaster" (the context being Lot fleeing the impending judgement and destruction of Sodom & Gomorrah).
KJV: Behold now, thy servant hath found grace in thy sight, and thou hast magnified thy mercy, which thou hast shewed unto me in saving my life; and I cannot escape to the mountain, lest some evil[H7451] take me, and I die:
NIV: Your servant has found favor in your eyes, and you have shown great kindness to me in sparing my life. But I can't flee to the mountains; this disaster[H7451] will overtake me, and I'll die.
I hope this points out that "evil" does not equal sin in all cases, especially in older translations/versions of the Bible. And in particular, "evil" does not equal sin in Isaiah 45:7, and the Bible does not say that God created sin.
But If God Created All Things Doesn't That Include Evil?
For a non-doctrinal answer (at least as far as possible), first consider the definition of evil. What is evil? If we are talking about evil as sin, then we can define evil to be unrighteousness.
Evil (by Biblical definition of sin) is unrighteousness, the absence of righteousness, just as darkness is the absence of light. Darkness and evil are not anything, and cannot be created, they are by definition the absence of their "opposites". Therefore, God did not create evil (sin and unrighteousness), evil simply exists wherever God is absent (not followed).
For further reading check: What is the definition of evil?
Why Does God Allow Evil?
The answers to this differ fundamentally based on doctrinal beliefs, mainly dependent on the view of free will.
Assuming that we have free will, evil (unrighteousness) is a byproduct of our free will whenever we choose to do something contrary to the nature of God.
So the question then becomes "why do we have free will?" This is yet another source of doctrinal differences, but one common reason is because when we choose to do God's will, we give honor and glory to God. This honor is all the more meaningful because it was our choice, we were not compelled like a marionette puppet to do it. Furthermore, without the option of unrighteousness and sin, there would be no point to Christ's sacrifice, and no need for forgiveness and redemption.