Yes, God did put within each of us a sense of right and wrong -- what we usually call our conscience. In fact, one reason we feel guilty when we do wrong is because our conscience is speaking to us. The Bible urges us to pursue things that come "from a pure heart and a good conscience and a sincere faith" (1 Timothy 1:5).
The problem is that our conscience -- that inner sense of right and wrong -- has been corrupted by sin. Have you ever known someone who had been burned over some part of their body, and their nerves had been damaged so much that they no longer had any feeling there? That's what sin does to our conscience; sin can scar it so much that we're no longer as sensitive to right and wrong as we should be. The Bible speaks of evil men "whose consciences have been seared as with a hot iron" (1 Timothy 4:2).
Our conscience is important; it still can alert us to right and wrong. But it's not always a dependable guide -- and it can even deceive us if we aren't careful. That's why we need Christ, who comes to live within us by His Spirit when we welcome Him into our lives. It's also why we need the Bible, which God has given to guide us.
Evil is powerful, and sometimes it overwhelms the conscience of someone who is given over to it. But God is more powerful, and He will guide us by His Word.