It is often debated whether John wrote primarily for unbelievers or for believers. The content of the Gospel itself leads me to believe that John has in mind primarily those who consider themselves to be believers, simply because much of what he says assumes the knowledge that believers would have. Yet this does not mean that his Gospel was not addressed to unbelievers, as well.
I believe that this debate sometimes suffers from a false dichotomy. It is not the case at all that these two purposes are in conflict with each other or even that these are two distinct purposes. The tension we feel may come more from our false pre-understanding that evangelism is somehow separate from discipleship, that believing is a momentary, one-time experience, or that the gospel has nothing important to say to believers. In reality, any true presentation of Jesus will shine on all men and force the issue of who he is and how they must respond to him. The Gospel of John calls upon everyone who hears, whether professing believer or unbeliever, to believe that Jesus is the Christ.