The short answer to this question is that the Bible does not speak to distinguishing between demon possession and a psychological disorder. Because God chose not to equip Christians for this task, we should probably assume this is not something we are called to do. However, there are two things we know for sure from Scripture.
First, we know from the Bible that demons can and do possess those who do not belong to Christ, and Scripture gives some examples of people being possessed by demons. From these descriptions, we can find some symptoms of demonic influence as well as gain insights as to how a demon possesses someone. In some of these passages, the demon possession causes physical ailments (inability to speak, epileptic symptoms, blindness, etc. [Matthew 9:32-33, Mark 9:17-18]); in other cases the demon causes the individual to do evil (Judas is the main example); in Acts 16:16-18, the spirit apparently gave a slave girl some ability to know things beyond her own learning (a spirit of divination); in the case of the demoniac of the Gadarenes who was possessed by a multitude of demons, he had superhuman strength, cut himself, went around naked, and lived among the tombstones (Mark 5:1-17). King Saul, after rebelling against the LORD, was troubled by an evil spirit (1 Samuel 16:14-15; 18:10-11; 19:9-10) with the apparent effect of a melancholy mood and an increased desire and readiness to kill David (God’s next anointed king of Israel).
Second, we should be greatly encouraged to know that it is impossible for a Christian to be possessed by a demon. The believer is indwelt by the Holy Spirit of God who comes to reside in our hearts when we give our lives to Christ (2 Corinthians 1:22). The description of a possessing demon as “evil” means “unclean” in the Greek (Mark 5:2), therefore making it impossible for the Holy Spirit to share His dwelling place with such a creature. For those who do not have the Holy Spirit, however, no amount of “cleaning up their lives” will prevent a demon from possessing or influencing them. The parable Jesus told in Matthew 12:43 makes this clear:
“When an unclean spirit goes out of a man, he goes through dry places, seeking rest, and finds none. Then he says, ‘I will return to my house from which I came.’ And when he comes, he finds it empty, swept, and put in order. Then he goes and takes with him seven other spirits more wicked than himself, and they enter and dwell there; and the last state of that man is worse than the first.” Jesus is teaching here that we are incapable of sweeping and putting “in order” our own hearts because our hearts are “deceitful above all things and desperately wicked” (Jeremiah 17:9 KJV). Only God can regenerate us and create in us a new heart (Ezekiel 36:26) and make us new creatures in Christ (2 Corinthians 5:17).
However, while Christians cannot be possessed by a demon, this is not to be confused with being influenced by one. The apostle Peter is an illustration of a believer who was influenced by the devil (Matthew 16:23). This can occur especially when we are not mature in the faith and not sufficiently involved in the spiritual disciplines of regular study of Scripture and prayer.
Lastly, a word to the wise: some people develop an unhealthy fascination with the occult and demonic activity. This is ill advised to say the least. If we pursue God with our lives and we are clothing ourselves with His armor and relying upon His strength (not our own) (Ephesians 6:10-18), we have nothing to fear from the evil ones, for God rules over all! When we are in Christ, and He in us, we have nothing to fear from the evil one because “He who is in you is greater than he who is in the world” (1 John 4:4).