Question: In the Old Testament there is not a sacrifice for deliberate sin or sins done intentionally. How did this person get back to the right relationship with God? Also the same in the New Testament does one re-crucify the Lord?
Response: In regard to "re-crucifying the Lord", I believe that you are referring to Hebrews 6:6 "crucifying the Son of God afresh". In that passage, Paul is stating the impossibility of recovery from sin and apostasy for the believers in Judea if they continue to participate in Jewish sacrificial ritual, because by doing so they were, in effect, saying that Christ's sacrifice had been ineffective (cf. Heb.10:29 which expresses the same sentiment in other words). It is true that the Mosaic Law has no sacrifice for "intentional sin" as defined by the law, because intentional violation of the law was to be punished with death (cf. Lev.24:10-23; Heb.10:26-28). Now the Law does not deal with every aspect of sin, although the tenth commandment covers a good deal of territory (it was this commandment, after all, that convinced Paul that keeping the Law was an impossibility: Rom.7:8). Much of what we say, think, and do can be sinful even though not specifically delineated as such in the Bible (cf. Jas.4:17). The ritual sacrifices of the Mosaic Law were impressive lessons to pre-Christ Israel of two critically important facts: 1) even the process of atoning for their sins of ignorance alone was not an easy one - how much more guilty and accountable then are we human beings for all the manifold sins we willingly and knowingly undertake? 2) complete atonement through blood sacrifice is clearly something impossible for man to achieve - therefore God will have to provide a Lamb for Himself (cf. Gen.22:8). In short, the blood sacrifice procedures of the Mosaic Law could never give the participants a consciousness of innocence - indeed, the entire point was to drive home their consciousness of sin and guilt, and so to make abundantly clear their need for God's deliverance from sin (cf. Heb.10:1-4). The Israelites, especially those who were following God, knew they were sinners:
There is no man who does not sin.
1st Kings 8:46 (cf. Ps.130:3)
Now to the underlying point of your question, what did they do about personal sin? God's requirements have always been the same. He wants us to follow Him, to worship Him, to obey Him from the heart. When we stray from Him, ignore Him, disobey Him, He disciplines us like a father does his sons. When we return to Him, acknowledge our sins before Him and give ourselves up to His will again, He takes us back, forgives us our sins, and bestows upon us once more all the joys of salvation.
This is the one I esteem, he who is humble and contrite in spirit and trembles at My Word. But whoever [just] sacrifices a bull is like one who kills a man and whoever [just] offers a lamb is like one who breaks a dog's neck.
Isaiah 66:2b-3a NIV
Search the Psalms and you will see (esp. Ps.32; Ps.51; Ps.57; Ps.130; Ps.143; cf. also Is.1:16-20) - gratitude for God's merciful forgiveness of sin (based upon a contrite prayer of confession) is everywhere, and there is no sense that such mercy is to be taken for granted because of animal sacrifice:
You do not delight in sacrifice, or I would bring it; you do not take pleasure in burnt offerings. The sacrifices of God are a broken spirit; a broken and contrite heart, O God, You will not despise.
Psalm 51:16-17 NIV
The solution to the problem of personal sin has always been the same. Tell God (confess). Get off the wrong road (have a change of heart about your sin). Then get moving on God's road again (obedient spiritual growth and service).
This and most other biblical aspects of sin are covered in detail in the following links:
Bible Basics 3B: Hamartiology: The Biblical Study of Sin
Lesson #15 of the Peter series: John's Primer on Sin
Hope this helps,
Yours in Christ,