As correctly explained in Jon's answer, non-belief in the Bible is normally in terms of practical atheism ("there is no god that impacts my life") rather than absolute atheism ("there is no higher power of any kind"). With that is mind, there are a large number of Bible verse that talk about atheism.
As the OP notes, Psalms 14:1 and the parallel in 53:1 are probably the clearest statements about unbelief.
The fool says in his heart, “There is no God.” They are corrupt, they do abominable deeds, there is none who does good. (14:1, ESV)
The fool says in his heart, “There is no God.” They are corrupt, doing abominable iniquity; there is none who does good. (53:1)
From these verses, it is clear that the Psalmist is concerned with practical atheism - the kind of belief that causes men to do bad things. Whether the "fool" believes that no God exists or only that no personal God exists, is not relevant. The result is the same either way - the person pursues selfish goals.
This theme is also expressed in Psalms 10:3-4,
For the wicked boasts of the desires of his soul, and the one greedy for gain curses and renounces the LORD. In the pride of his face the wicked does not seek him; all his thoughts are, “There is no God.”
Psalms 36:1-2,
Transgression speaks to the wicked deep in his heart; there is no fear of God before his eyes. For he flatters himself in his own eyes that his iniquity cannot be found out and hated.
and Proverbs 11:9
With his mouth the godless man destroys his neighbor, But through knowledge the righteous will be delivered.
Essentially, "atheism" to these authors is equivalent to believing yourself to be the ultimate authority. I.E., he is describing those who do not feel they need to answer to a higher power for their actions.
Proverbs 30:8-9a is especially relevant, I think:
Remove far from me falsehood and lying; give me neither poverty nor riches; feed me with the food that is needful for me, lest I be full and deny you and say, “Who is the LORD?”
In this prayer, Agur asks not to be too satisfied with his life lest he turn deny the need for God. By asking "Who is the LORD?" the "full" man is basically saying "God doesn't exist because I don't need Him."
In the New Testament, John puts it plainly:
Everyone who goes on ahead and does not abide in the teaching of Christ, does not have God. (2 John 1:9)
Paul takes this idea a step further, arguing that all should know about the existence and goodness of God. It is only through ignorance caused by blinding sin that some remain unbelievers.
They are darkened in their understanding, alienated from the life of God because of the ignorance that is in them, due to their hardness of heart. (Ephesians 4:18)
For what can be known about God is plain to them, because God has shown it to them. For his invisible attributes, namely, his eternal power and divine nature, have been clearly perceived, ever since the creation of the world, in the things that have been made. So they are without excuse. For although they knew God, they did not honor him as God or give thanks to him, but they became futile in their thinking, and their foolish hearts were darkened. Claiming to be wise, they became fools (Romans 1:19-22)
And even if our gospel is veiled, it is veiled to those who are perishing. In their case the god of this world has blinded the minds of the unbelievers, to keep them from seeing the light of the gospel of the glory of Christ, who is the image of God. (2 Corinthians 4:3-4)
He also warns against the knowledge of man that blinds a person to the truth of God's word:
O Timothy, guard the deposit entrusted to you. Avoid the irreverent babble and contradictions of what is falsely called “knowledge,” for by professing it some have swerved from the faith. (1 Timothy 6:20-21)
See to it that no one takes you captive by philosophy and empty deceit, according to human tradition, according to the elemental spirits of the world, and not according to Christ. (Colossians 2:8)
Those who do not know the true God inevitably worship false gods - perhaps literally, most equally as ungodly by making false "gods" out of whatever is around. See especially the Golden Calf story in Exodus 32 and also:
Formerly, when you did not know God, you were enslaved to those that by nature are not gods. (Galatians 4:8)
But where are your gods that you made for yourself? Let them arise, if they can save you, in your time of trouble; for as many as your cities are your gods (Jeremiah 2:28)
Or, by following the oldest temptation, making oneself a "god". See the Fall in Genesis 3 where the serpent promises Eve "you will be like God".
##Conclusion##
So, the Bible does speak of atheists in many places. However, it is generally unconcerned with whether one believes in the existence of a god. Rather, one must act as if they believe in the true, personal God to not be equated with atheism. As such, anyone who denies Yahweh as the true God is an atheist from a Biblical perspective. Certainly, there are many self-professed "atheists" of this variety in the Bible. There are not, however, unambiguous statements of belief in the non-existence of God, because this is not a distinct thing from belief in the uselessness of God, according to the Bible.
