This might seem like a perplexing question, but it does have an answer. Though the Christian who has committed suicide has committed a grave sin, he is still forgiven. But, in order to understand why a Christian who commits suicide is forgiven, we first need to understand what salvation is and what it is based upon.
Salvation is the state of being saved from God's judgment upon the sinner. The only way to be saved is to trust Jesus for the forgiveness of one's sins (John 14:6, Acts 4:12). All who do not trust Jesus alone, by faith (Rom. 5:1; Rom. 6:23; Eph. 2:8-9) are not forgiven and go to hell when they die (Matt. 25:46; John 3:18). When Jesus forgives someone, He forgives all their sins and gives them eternal life; and they shall never perish (John 10:28). He does not give them temporary eternal life--otherwise, it would not be eternal.
Salvation is not based upon what you do. In other words, you don't have to obey any Law of God in order to become saved. This is because no one is saved by keeping the Law of God (Gal. 2:21; Rom. 3:24-28). But that does not mean that you can go and sin all you want. Rom. 6:1-3 expressly condemns such action. Instead, we are saved for the purpose of purity (1 Thess. 4:7). Our salvation is strictly from God: "By grace through faith you have been saved . . . " (Eph. 2:8). Other than acting by faith in trusting and accepting what Jesus did on the cross, you don't do a thing (John 1:12-3) in order to become saved. Since you did not get your salvation by what you did, you cannot lose it by what you do.
What about the unforgivable sin? Is that suicide? No. Suicide is not the unforgivable sin. Jesus spoke of the unforgivable sin in Matt. 12:22-32. The context is when the Pharisees accused Jesus of casting out demons by the power of the devil. Therefore, suicide is not the unforgivable sin.
Is repentance necessary for salvation?
This is a good question, and the answer is yes--and no. Now, before you throw stones, hear me out. Repentance is a necessary result of the saving work of God--not the cause of salvation. If repentance brought salvation, then salvation is by works or rather, the ceasing of bad works. That isn't how it works. God grants repentance to the Christian (2 Tim. 2:25). The Christian then turns from his sin; that is, he stops sinning. He is able to repent because he is saved--not to get saved.
In 1 John 1:9 it says, "If we confess our sins He is faithful and just to forgive us our sins and cleanse us from all unrighteousness." Confession of sin and its natural result of repentance are necessary elements of the Christian's life. But, what about the sins that we do not know we commit? If we do not confess them and do not repent of them, are we still saved? Of course we are! Otherwise, we would be forced to confess and repent of every single sin we ever commit. In effect, we'd be back under the Law--living by a rule of absolute repentance of every detail lest you be damned. This is bondage--not freedom. Jesus said His yoke was light--not hard (Matt. 11:27-30.
So, repentance is not the cause of salvation, but it is a result of salvation. The believer repents from his sins upon trusting in Christ and thereafter continues to repent of further sins that the Lord reveals to him.
Back to the suicide issue
Suicide is, in effect, self-murder. The unfortunate thing about it is that the one who commits it cannot repent of it. The damage is permanently done. We can see in the Bible that murderers have been redeemed (Moses, David, etc.,), but they had opportunities to confess their sins and repent. With suicide, the person does not. But that does not mean the person is lost. Jesus bore all that person's sins--including suicide. If Jesus bore that person's sins on the cross 2000 years ago and if suicide was not covered, then the Christian was never saved in the first place; and the one sin of suicide is able to undo the entire work of the cross of Christ. This cannot be. Jesus either saves completely, or he does not.
Is suicide always wrong?
That I cannot answer because I cannot list every possible situation. But, it seems obvious that suicide is clearly wrong though forgivable. However, there are general categories of suicide on which we could briefly comment:
Medically Assisted Suicide--I've never seen this as being acceptable. The doctor is supposed to save life--not destroy it. But, lately as destroying the lives of the unborn is more common place, destroying the lives of the sick has become the next logical step.
Suicide to prevent prolonged torture--Let's say that someone was being tortured in an excruciating manner for an unbearably long period of time, is suicide an option? Perhaps. But if it were in this situation, why wouldn't it be all right in the medically-assisted context if the patient were also in excruciating pain for long periods of time? Quite honestly, I'm not sure how to answer that one.
Suicide due to depression--Of course, this is never a good reason for suicide. Seasons pass and so does depression. The one who is depressed needs to look to Jesus and get help. Depression is real and powerful and is best fought with help. Also, severe depression robs the mind of clear thinking. People in such states are despondent--not in their right mind.
Suicide due to a chemical imbalance in the brain--The human brain is incredibly complex, and the medical community is full of accounts of extraordinary behaviors by people whose "circuits got crossed." I don't see how a situation like this would make it justifiable. I think it simply would make it more explainable.
Accidental suicide--Sometimes people accidentally kill themselves. This could mean leaning over a balcony too far and falling to one's death or actually purposefully taking a stupid risk like playing with a gun. Of course, with either, stupidity does not remove us from the grace of God.
Is the Christian forgiven for suicide? Yes. But suicide is not an option. We do not have the right to take our own lives. That belongs to God.