A working definition of the Trinity is one God subsists in three persons, Father, Son, and Holy Spirit, co-equal and co-eternal. My contention is the Bible supports the following three premises to establish the Trinity: 1) there is only one God, 2) the Father, Son, and Holy Spirit are separate persons, and 3) each person is fully God. If all three of these premises are true then the only rational conclusion is the Bible teaches the doctrine of the Trinity.
1. The Bible teaches there only one God.
This is actually a very easy task because many verses support this premise. There are minimally 28 verses in the New and Old Testaments that clearly teach there is only one God. I will provide a few straightforward examples.
Deuteronomy 6:4, “Hear (Shema) O Israel: The Lord our God, the Lord is one.”
Isaiah 44:6, 8 “Thus says the Lord, the King of Israel and his Redeemer, the Lord of hosts: ‘I am the first and I am the last, and there is no God besides Me.  ‘Do not tremble and do not be afraid; have I not long since announced it to you and declared it? and you are My witnesses. Is there any God besides Me, or is there any other Rock? I know of none.'”
Isaiah 45:5a, “I am the Lord, and there is no other; apart from me there is no God.”
Mark 12:29-34 “Jesus answered, ‘The foremost is, ‘Hear, O Israel! The Lord our God is one Lord;  and you shall love the Lord your God with all your heart, and with all your soul, and with all your mind, and with all your strength.'  ‘The second is this, ‘You shall love your neighbor as yourself.' There is no other commandment greater than these.’  And the scribe said to Him, ‘Right, Teacher, You have truly stated that He is One; and there is no one else besides Him;  and to love Him with all the heart and with all the understanding and with all the strength, and to love one's neighbor as himself, is much more than all burnt offerings and sacrifices.’  And when Jesus saw that he had answered intelligently, He said to him, ‘You are not far from the kingdom of God.’ And after that, no one would venture to ask Him any more questions.”
Here are some additional verses from the Bible that teach there is only 1 God. Deut. 4:35, 4:39, Is. 45:14, 45:21, 45:22, 46:9, Joel 2:27, James 2:19, and John 17:3. Christianity historically has always been considered a monotheistic religion and condemns polytheism. Premise one is established, the Bible teaches there is only one God.
2. The Bible Teaches the Father, Son, and Holy Spirit are separate persons.
A person is a living being who is non-physical, hence immaterial. A person can love, be rational, communicate, and have emotions. It is easy to use the Bible to establish the personhood of the Father and the Son. The scriptures record how the Son walked the earth as a human person. John 3:16 states the Father loved the world when He gave His one and only Son. Only a person can choose to love. Many other Biblical examples could be supplied to demonstrate the Father as a personal being.
It is a little more difficult to show the Holy Spirit as a person. A passage that demonstrates the personhood of the Holy Spirit can be found in John 14. Here we see Jesus speaking to his disciples during the last supper. John 14:25-26, “All this I have spoken while still with you. But the Counselor, the Holy Spirit, whom the Father will send in my name, will teach you all things and will remind you of everything I have said to you.”
These verses support the Holy Spirit as a person. Only a person can be a counselor, teach people, and remind others. In John chapters 14-16 we see actions performed by the Father, the Son, and the Holy Spirit that clearly shows how each is a person.
Another example can be found in Acts 13:2. “While they were worshiping the Lord and fasting, the Holy Spirit said, ‘Set apart for me Barnabas and Saul for the work to which I have called them.’” Here the Holy Spirit speaks and commands Barnabas and Saul to be set apart for the work that the Holy Spirit has called them. Speaking, commanding, and setting forth a plan can only be done by a person. Many more scriptures give evidence for the personhood of the Holy Spirit.
The third and final premise is all three are God. The Bible teaches this truth while at the same time maintaining the first premise there is only one God. For most critics of the Trinity this is where the real battle begins.