Do any people in the Bible self identify as atheists?
Or are they all theists, differing in which god they follow?
Apparently at least some atheists existed, since David had heard of them:
The fool says in his heart, “There is no God.”
Psalm 14:1 and Psalm 53:1.
Not exactly the answer you are looking for, but Paul clearly understands humanity to be functionally atheistic, if not in practice. From Romans 1:
18 The wrath of God is being revealed from heaven against all the godlessness and wickedness of people, who suppress the truth by their wickedness, 19 since what may be known about God is plain to them, because God has made it plain to them. 20 For since the creation of the world God’s invisible qualities—his eternal power and divine nature—have been clearly seen, being understood from what has been made, so that people are without excuse. 21 For although they knew God, they neither glorified him as God nor gave thanks to him, but their thinking became futile and their foolish hearts were darkened. 22 Although they claimed to be wise, they became fools 23 and exchanged the glory of the immortal God for images made to look like a mortal human being and birds and animals and reptiles. 24 Therefore God gave them over in the sinful desires of their hearts to sexual impurity for the degrading of their bodies with one another. 25 They exchanged the truth about God for a lie, and worshiped and served created things rather than the Creator—who is forever praised. Amen.
In practice, to be a-theistic in the ancient world was to invite destruction upon the whole community. In Rome, in particular, for example, legally they didn't care which god you worshipped, as long as you worshipped one. I don't have the quote handy, but I remember David Bell in Many Mansions making the case that atheism was like "poor hygenie or a public health hazard," meaning that impiety would invite needless retribution from someone, so if everybody was worshipping what they thought, at the least the "true gods," would forgive them.
King Sennacherip and his general Rabshakeh come closest to being self-proclaimed atheists in the Bible. Rabshakeh says in the name of the Gentile king to King Hezekiah:
Where are the gods of Hamath, and of Arpad? where are the gods of Sepharvaim, Hena, and Ivah? have they delivered Samaria out of mine hand? Who are they among all the gods of the countries, that have delivered their country out of mine hand, that the LORD should deliver Jerusalem out of mine hand?
Here Sennacherib denounces all supernatural intervention to save, not just of Yahweh, but of the dozens of Canaanite gods around Israel. He essentially is saying that the gods spoken of here, are not gods.
Besides this example, I must conclude that the answer to your question is no. There are no self-identified atheists in the Bible.
The NET Bible notes:
“There is no God.” The statement is probably not a philosophical assertion that God does not exist, but rather a confident affirmation that God is unconcerned about how men live morally and ethically (see Ps 10:4, 11).
and:
Heb “they act corruptly, they do evil [with] injustice.” Ps 14:1 has עֲלִילָה (’alilah, “a deed”) instead of עָוֶל (’aval, “injustice”). The verbs describe the typical behavior of the wicked. The subject of the plural verbs is “sons of man” (v. 2). The entire human race is characterized by sinful behavior. This practical atheism—living as if there is no God who will hold them accountable for their actions—makes them fools, for one of the earmarks of folly is to fail to anticipate the long range consequences of one’s behavior.
If you read through the text of these nearly identical Psalms, you will notice they concentrate almost entirely on moral foolishness rather than any sort of intellectual foolishness. The phrase "There is no God" seems more like hyperbole than a statement of fact. Nearly everyone in the pre-modern word would have been classified as theist in the broadest sense. A perhaps more representative verse would be:
Have they no knowledge, all the evildoers who eat up my people as they eat bread and do not call upon the LORD?—Psalm 14:4 (ESV)
My interactions with atheists (both online and off) suggest that this particular Psalm doesn't apply to everyone who says there is no God. Many people who claim the title "atheist" believe their actions have consequences. They do their best to raise their children lovingly and ethically. They support worthy causes and give of themselves to make the world a better place. They refrain from wicked deeds and mistreating others. In fact, this Psalm applies much better to certain people I know who claim to love God, but by their actions show they don't believe He exists.
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".
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.
In addition to the other excellent answers, it should also be noted that the Epicurean philosophers, mentioned in Acts 17:18, were in essence atheists. They did not believe in God or gods who intervened in the physical universe. In fact, the problem-of-evil argument against belief in God was apparently first formulated by Epicurus himself, in the form of the Epicurean paradox, which asserts that the three traits attributed to God, namely omnipotence, omniscience, and goodness, cannot all be true without contradiction:
In fact, in many particulars, Epicureanism was indistinguishable from modern atheism. In addition to their disbelief in any interventionist gods, they also were materialists who believed that everything that happens is a result of the material properties of the atoms out of which everything is made. They believed in a version of the multiple universes hypothesis to explain the appearance of design in the universe. They disbelieved in an immaterial human soul or continued existence after death. They believed that the good life is achieved by maximizing pleasure in this life (though contrary to the stereotype, they were not base hedonists, as Epicurus knew that such a lifestyle would not lead to happiness in the long term.) They believed that one ought to not harm others because this ethic maximizes the ability of each person in a society to live a pleasurable and pain-free life.
