What does the Bible say?
The act of suicide is sin, because it's taking something that is strictly under God's commandments (life and death) and subjecting it to our own will. It robs a person of the amazing gift of life given to them by God alone.
"For you created my inmost being; you knit me together in my mother’s womb." — Psalm 139:13
More importantly the mental attitude and thought process of someone who commits suicide is certainly sinful. It is always based in a selfish thought process.
This is considered sinful because it is the purest form of unbelief in the promises of God. The general thought process of a suicidal person recognises that they do not believe God can help them. They believe no one can help them and that they are past the point of no return.
Does a true child of God show fruits of suicide? God certainly has mercies on those who can't make choices for themselves, but ultimately, those with a self-destructive behaviour or a debased mind as described in Romans 1:28 are ultimately choosing to service themselves over God.
Note that no one has ever hated his own flesh1, such behaviour reveals that those who are suicidal have no faith in Christ. God has made Himself clear that the unrighteous will not inherit the kingdom of God2, and that the unbelieving go to the lake of fire3.
Biblical precedent
Suicide is only mentioned in the Bible six times. In the case of Abimelech4, Samson5, Saul6, Ahithophel7, Zimri8, and Judas Iscariot9. Each of these examples can be individually assessed and investigated as to how they speak about suicide through the eyes of the Father.
Works performed outside of God are sin. If you believe this to be true, the deaths depicted in these texts can be quickly and easily explained as sinful, or as righteous (as we see in Samson's example, it's more the case of a noble sacrifice over suicide).
Nuance is required
Please note that by writing this I am not saying that all people who commit suicide aren't saved. God is ultimately the judge and He will judge each person accordingly.