While the doctrine of the Christian Trinity is not explicitly taught in any one single passage of the Bible, the concept is definitely present when you look at the Scriptures collectively and put everything together in harmony. 

The term Trinity, formulated by early Church Father Tertullian, is imperfect. It seems to emphasize only threeness. (I thought this on my own without it being suggested to me.) Theologians such as Princeton's B.B. Warfield suggest Triunity as a better term because it is inclusive of both unity and plurality. 

Traditionally and historically the Church has defined the Trinity (Triunity) as One God in three Persons. Again the term Persons is imperfect. We tend to think of three persons as distinct and acting independently of each other. But this is not what is meant. In the Triunity there is one Essence that is undivided. The Persons act in perfect mutual submissions to one another. 

So Jesus said to them, “Truly, truly, I say to you, the Son can do nothing of his own accord, but only what he sees the Father doing. For whatever the Father does, that the Son does likewise.
- John 5:19 

Historically Jews, Muslims, and Greco-Roman Pagans labeled Christians as polytheists, believing we worship three Gods. But this is not the case. Scripture plainly teaches that there is only one God: 

Hear, O Israel: The Lord our God, the Lord is one. - Deuteronomy 6:4 

And, 

For thus says the LORD,
who created the heavens
(he is God!),
who formed the earth and made it
(he established it;
he did not create it empty,
he formed it to be inhabited!):
I am the LORD, and there is no other. - Isaiah 45:18 

And, 

You believe that God is one; you do well. Even the demons believe—and shudder! - James 2:19 

There is only one God. And yet even in the Old Testament, Scripture allows for and even indicates that the one true God exists in plurality. This subtext as it were is actually present in the very first verse of the Bible. 

In the beginning, God created the heavens and the earth. - Genesis 1:1 

The Hebrew word for God is Elohim, which is the plural form of the word El, meaning God, Mighty One, or Power. In English we often make a word plural by adding ‘s’ to the ending. Similarly, the -im ending in Hebrew is one way to make a word plural (this is transliterated of course since Hebrew doesn't use Roman letters). An example is cherub and cherubim - the singular and plural form indicating an order of angel. 

Thus the word for God (Elohim) is actually plural. Yet the verb that follows - created (bara) - conjugates in the singular. Technically this is a grammatical error. Within the context of inspired, infallible Scripture this is a marker that we should take note of. 

Then God said, “Let us make man in our image, after our likeness. - Genesis 1:26 

Notice the use of the first person plural pronoun - ‘us’. This is another clear indication of plurality - again in the very first chapter of the Bible. The same occurs two more times in the Book of Genesis: 

Then the LORD God said, “Behold, the man has become like one of us in knowing good and evil. - Genesis 3:22 

And, 

Come, let us go down and there confuse their language, so that they may not understand one another’s speech. - Genesis 11:7 

We mentioned Deuteronomy 6:4 already, a foundational Jewish prayer known as the Shema. Let's look at it again. 

Hear, O Israel: The Lord our God, the Lord is one. 

The Hebrew word for one (echad) can imply a composite unity, as in a cluster of grapes - one cluster composed of many grapes. This of course is an imperfect analogy but again we find this concept of plurality in oneness embedded in the nuance of the Hebrew language. 

We find the Lord saying ‘us’ one more time in Isaiah 6:8: 

And I heard the voice of the Lord saying, “Whom shall I send, and who will go for us?” Then I said, “Here I am! Send me.” 

From what has been stated so far a full-fledged doctrine of Triunity cannot be formulated. But groundwork has been laid. Now let us turn to the New Testament, where that which is concealed in the Old Testament is revealed. 

Triunitarian Formulations in the New Testament
There are several verses in the New Testament that seem to use Triunitarian formulations. 

Go therefore and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit. - Matthew 28:19 

The grace of the Lord Jesus Christ and the love of God and the fellowship of the Holy Spirit be with you all. - 2 Corinthians 13:14 

For there are three that bear record in heaven, the Father, the Word, and the Holy Ghost: and these three are one. - 1 John 5:7 KJV 

…elect according to the foreknowledge of God the Father, in the sanctification of the Spirit, for obedience to Jesus Christ and for sprinkling with his blood. - 1 Peter 1:2 

Additionally, we see all three members of the Godhead present at the Baptism of Jesus: 

And when Jesus was baptized, immediately he went up from the water, and behold, the heavens were opened to him, and he saw the Spirit of God descending like a dove and coming to rest on him; and behold, a voice from heaven said, “This is my beloved Son, with whom I am well pleased.” - Matthew 3:16-17 

Jesus is baptized, the Holy Spirit comes and rests upon Jesus, and the Father speaks from Heaven.

Old Testament Allusions?
It has been suggested or assumed that there are no Trinitarian passages in the Old Testament. However, upon closer examination there may be at least two passages to consider. 

Some have suggested that Psalm 2 represents a conversation or dialogue within the Godhead. 

Verses 1-3 - the Holy Spirit
Verses 4-6 - the Father
Verses 7-9 - the Son
Verses 10-12 - the Holy Spirit 

Isaiah 48:16 is also the interesting: 

Come near to Me, hear this:
I have not spoken in secret from the beginning;
From the time that it was, I was there.
And now the Lord GOD and His Spirit
Have sent Me. (NKJV) 

The one speaking is the Son (the Word who was with God in the beginning. The Lord God (the Father and (the Holy) Spirit have sent Him. 

Regarding these last two Old Testament passages being possible allusions to the Triunity, I would not be dogmatic about it but both readings seem plausible to this author.