Can a murderer be forgiven by God and still go to heaven?
Jesus stated, "Assuredly, I say to you, all sins will be forgiven the sons of men, and whatever blasphemies they may utter; but he who blasphemes against the Holy Spirit never has forgiveness, but is subject to eternal condemnation" (Mark 3:28-29). Since only one sin is unforgivable and that sin is not murder; yes, it is possible for a murderer to be forgiven.
In fact, there are several cases of this happening in the Scriptures. David tried to cover up his adultery with Bathsheba by having her husband killed, thereby allowing him to marry his widow. However, God knew. "Why have you despised the commandment of the LORD, to do evil in His sight? You have killed Uriah the Hittite with the sword; you have taken his wife to be your wife, and have killed him with the sword of the people of Ammon" (II Samuel 12:9). David owned up to his sin and was forgiven, though he still had to face the consequences of his sin. "So David said to Nathan, "I have sinned against the LORD." And Nathan said to David, "The LORD also has put away your sin; you shall not die. However, because by this deed you have given great occasion to the enemies of the LORD to blaspheme, the child also who is born to you shall surely die" (II Samuel 12:13-14).
Another case is the apostle Paul. Before he became a Christian he was known by the name of Saul. As a young man he was involved in the murder of Stephen (Acts 7:57-8:1). He was a leading persecutor of early Christians. "As for Saul, he made havoc of the church, entering every house, and dragging off men and women, committing them to prison" (Acts 8:3). Paul described it in this manner, "This I also did in Jerusalem, and many of the saints I shut up in prison, having received authority from the chief priests; and when they were put to death, I cast my vote against them. And I punished them often in every synagogue and compelled them to blaspheme; and being exceedingly enraged against them, I persecuted them even to foreign cities" (Acts 26:10-11). And yet the wonder of it all is that Jesus sought out Saul and salvation was offered to this murderer of Christians. "I thank Christ Jesus our Lord who has enabled me, because He counted me faithful, putting me into the ministry, although I was formerly a blasphemer, a persecutor, and an insolent man; but I obtained mercy because I did it ignorantly in unbelief. And the grace of our Lord was exceedingly abundant, with faith and love which are in Christ Jesus. This is a faithful saying and worthy of all acceptance, that Christ Jesus came into the world to save sinners, of whom I am chief" (I Timothy 1:12-15). At the end of his life, Paul, the former murderer, said, "I have fought the good fight, I have finished the race, I have kept the faith. Finally, there is laid up for me the crown of righteousness, which the Lord, the righteous Judge, will give to me on that Day, and not to me only but also to all who have loved His appearing" (II Timothy 4:7-8).
How can this be? The answer is straight forward: God doesn't want anyone to perish. He is willing to forgive if they will turn away from their sins and follow His will. "Again, when a wicked man turns away from the wickedness which he committed, and does what is lawful and right, he preserves himself alive. Because he considers and turns away from all the transgressions which he committed, he shall surely live; he shall not die. ... Therefore I will judge you, O house of Israel, every one according to his ways," says the Lord GOD. "Repent, and turn from all your transgressions, so that iniquity will not be your ruin. Cast away from you all the transgressions which you have committed, and get yourselves a new heart and a new spirit. For why should you die, O house of Israel? For I have no pleasure in the death of one who dies," says the Lord GOD. "Therefore turn and live!"" (Ezekiel 18:27-28, 30-32).