Effectively, what Claudius is saying here is that he has not been praying in earnest, and therefore God will not hear his words. Claudius says that his "words fly up," as he offers them to heaven, but his "thoughts" remain below--there is a connotation in "below" in much of Shakespeare, where the word signifies not only the physical earth, but also the lower regions, or Hell; Claudius is ostensibly praying to God, but in his mind, he does not have Godly motivations. Claudius is aware, of course, that God will see through him and that his prayer will be ineffective, because insincere words--"words without thoughts" attached to them--will never "to heaven go." Therefore, the viewer sees that Claudius has been very lucky in having avoided Hamlet's wrath: Hamlet allowed his uncle to continue praying unhampered because he did not wish to cause Claudius's soul to be damned by killing him in the middle of prayer. The irony for the viewer, then, is that, although Claudius himself feels that his prayer has been insincere and believes it cannot have not reached God, the act of this insincere prayer may have saved him after all. 

These lines come after Claudius has been trying to pray after he sees the play "The Mousetrap." What he doesn't know is that Hamlet sees him praying and almost kills him during that time. Hamlet decides not to kill Claudius because, according to Elizabethan belief, a person killed in the middle of prayer and confession of sin would go directly to heaven. Hamlet knows his own father was killed when he was sleeping. His father said the murder meant that he was "sent to my account/With all my imperfections on my head." ( Act I,scene v, lines 83-84)In other words, he is not in heaven, but in purgatory waiting for his sins to be burned away.

So Hamlet decides not to kill Claudius because he wants to wait until Claudius is doing something sinful, then kill him so he will be sent to hell or purgatory. Then Hamlet leaves and Claudius says these lines,"“My words fly up, My thoughts remain below. Words without thoughts never to heaven go” In other words, he is not willing to repent of his sin, therefore, the sin will not be forgiven. So Claudius' words are flying up but because he does not have the repentance to go with the words, his words will never reach heaven. Ironically, Claudius is in an unrepentant state and if Hamlet would have killed him, Claudius would not have been set to heaven, but the hell or purgatory as he wished.

What technique is used in the phrase "my words fly up, my thoughts remain below. Words without thoughts never to heaven go"?

Although this could be considered an example of personification, the lines within context also represent situational irony.

These lines fall within Act 3 Scene 3 of Hamlet and at this point, Claudius has just finished viewing the play within a play that Hamlet has devised to determine his guilt. Overcome with emotion, Claudius is found praying for forgiveness. Hamlet finally has both the truth (the ghost was right and Claudius murdered Hamlet's father) and the opportunity (Claudius is alone) for his revenge. However, he decides that he cannot kill Claudius at this moment because according to his beliefs, people killed while praying go straight to heaven. So once again, Hamlet delays and leaves Claudius alive.

When Hamlet exits, Claudius delivers these lines, and the audience learns that Claudius wasn't effectively praying at all. His words had no repentance behind them; he didn't mean what he was saying. This is similar to the saying that one can "feel his prayers bouncing off the ceiling."

Because Claudius isn't repentant, Hamlet could have followed through with his plans at this moment and killed Claudius. Nothing is standing in his way, after all. Since he doesn't realize this, the lines create situational irony.

What technique is used in the phrase "my words fly up, my thoughts remain below. Words without thoughts never to heaven go"?

This particular metaphorical device is called “personification,” where an abstraction is given physical properties to compare the abstraction with a visible, imaginable action. “Just like a bird flies up to the sky, my words fly up to heaven.  But because the words are hollow and empty, the thoughts of the speaker do not fly up, but remain on earth and are not received (by a Deity).”  It is sometimes called an “animation”; technically, a personification must give human qualities to an inanimate object:  “The sun smiled at me today.” These words, a rhyming couplet that gives closure to Hamlet’s soliloquy (another example of Shakespeare’s creative genius, to have someone else’s lines close Hamlet’s soliloquy), are spoken after the king’s prayers, outwardly of penitence.  He is praying as Hamlet watches him and decides not to kill him while he prays, because the King’s soul would go to heaven.  But the King rises, realizing that the insincerity of his prayers makes them useless—he still is greedy for power and is not really remorseful.