First, I would quote from the Hebrew scriptures in the Tanakh:
“You guide me with your counsel, and afterwards you will take me into glory.” (Psalm 73:24)
“But God will redeem my life from the grave; he will surely take me to himself.” (Psalm 49:15)
"The days of our years are threescore and ten...or if fourscore yet their strength is labour and sorrow, for it is soon cut off and away we fly." (Psalm 90:10)
“Do not put your trust in princes, in mortal men who cannot save. When their spirit departs they return to the ground” (Psalm 146:4)
“Remember God… before the dust returns to the ground it came from, and the spirit returns to God who gave it… For God will bring every deed into judgment, including every hidden thing, whether it is good or evil.” (Ecclesiastes 12:1-14)
“Multitudes who sleep in the dust of the earth will awake, some to everlasting life, others to shame and everlasting contempt… You will rest, and then at the end of the days you will rise to receive your allotted inheritance.” (Daniel 12:2-13)
“I know that my Redeemer lives, and that in the end he will stand upon the earth. And after my skin has been destroyed, yet in my flesh I will see God; I myself will see him with my own eyes.” (Job 19:25-26)
And don’t forget Enoch, for God took him instead of allowing him to die, and Elijah, who was caught up to heaven in a whirlwind without dying either – Hebrews 11:5.
The idea of going to heaven is not developed in these texts, but belief in the resurrection of the dead certainly is mentioned, and given that all those writers took God to be in heaven, whenever they spoke of their spirits departing, and “away we fly” etc, they had God’s presence in mind, which would be heaven. However, this was intermingled with belief in a future day of resurrection and did not rule out earth as a future home either. This is complicated, but knowing what the Jews believed about this helps clarify.
Ancient Hebrew scriptures had two words, qe’ver (grave) and sheol (to which the grave was the entrance). Sheol is often translated as hell. The Rabbinic schools of Shammai and Hillel taught that sheol had two compartments, hell and ‘the Bosom of Abraham’, with a great chasm separating them. This holding place was for the departed spirits of the dead, but on the Day of the Resurrection, they would come forth to be resurrected and judged. Jesus, a Jewish rabbi, agreed with those aspects of Judaic belief as can be seen from His description of hell, and the bosom of Abraham, in the warnings He gave to lovers of money (for the rich man who died ended up in agonizing torment in hell). See Luke 16:13-31.
See “The Life And Times of Jesus The Messiah” by Alfred Edersheim (1971) Appendix XIX, On Eternal Punishment, according to the Rabbis and the New Testament” (see vol. II Book V ch. Vi) http://www.ccel.org/ccel/e “Josephus’ Discourse to the Greeks Concerning Hades” as in “The Works of Josephus” translated by William Whiston, 1980, page 637 http://www.ccel.org/search/fulltext/Josephus%27%20Discourse%20to%20the%20Greeks%20Concerning%20Hades http://www.middletownbiblechurch.org/doctrine/hades.htm
When Jesus came, He spoke much more about heaven, and that is when the doctrine of believers going to heaven developed fully. He himself told his followers that he would return to heaven (after his resurrection) to go and prepare a place for them in his Father’s house (heaven). Then Jesus would return to fetch them so they could be with him and see his glory. That was in heaven. John 14:1-28 (“I said I go unto the Father” – who is in heaven.) Now here is a list of New Testament scriptures that assure believers in Jesus of their calling to be with Christ in heaven.
Matthew 5:12 -because great is your reward in heaven
Matthew 7:21 -those doing the will of my Father will enter the kingdom of heaven
Matthew 19:14 -the kingdom of heaven belongs to such ones
Matthew 19:21 -and you will have treasures in heaven
Luke 10:20 -your names are written in heaven
John 14:2-3 “In my Father’s house are many mansions: if it were not so, I would have told you. I go to prepare a place for you. And if I go and prepare a place for you, I will come again and receive you unto myself; that where I am, there ye may be also.”
John 17:24 “Father, I want those you have given me to be with me where I am and to see my glory, the glory you have given me because you loved me before the creation of the world.”
1 Corinthians 15:46-53 -as is the man from heaven [Jesus] so also are those who are of heaven And just as we have borne the likeness of the earthly man, so shall we bear the likeness of the man from heaven... we will all be changed - in a flash, in the twinkling of an eye, at the last trumpet... the dead will be raised imperishable and we will be changed...
2 Corinthians 5:1-2 -an eternal house in heaven, not built by hands... to be clothed with our heavenly dwelling
Romans 8:15-17 -co-heirs with Christ... in order that we may also share in his glory
Philippians 1:20-24 & 27 -I desire to depart [in death] and be with Christ, which is better by far [for he is in heaven, and so will I be once I die!]
Phil. 3:14 -I press on towards the goal to win the prize for which God has called me heavenwards in Christ
Philippians 3:20 -But our citizenship is in heaven
2 Timothy 4:18 -bring me safely to his heavenly kingdom
Hebrews 11:5 – “By faith Enoch was taken up to heaven without dying: he disappeared because God took him’
Verses 10 to 16 – “Abraham was confidently looking forward to a city with eternal foundations, a city designed and built by God… But they were looking for a better place, a heavenly home.”
Hebrews 12:22 – Christians come to Mount Zion, the city of God, heavenly Jerusalem where countless angels are gathered [not earthly Jerusalem].
Verse 23 – “You have come to the assembly of God’s firstborn children whose names are written in heaven. You have come to God himself, who is the judge over all things. You have come to the spirits of the righteous ones in heaven who have now been made perfect. You have come to Jesus [who is in heaven]…”
Revelation 3:21 where Jesus says, “To him who overcomes, I will give the right to sit with me on my throne, just as I overcame and sat down with my Father on his throne.” The throne Jesus speaks of is in heaven – see Rev. chapter 5.
Revelation 7:9-11 where a great multitude of people who have come out of the great tribulation on Earth are standing before God and Christ sitting on the throne of heaven, worshipping.
Revelation 6:9-11 -under [heaven’s] altar I saw the souls of those who had been slain… for their testimony
Revelation 19:1 -I saw a great multitude in heaven shouting, Salvation we owe to the Lamb...
Not all of those scriptures actually say believers in the old covenant and believers in Jesus (who are in the new covenant) will be in heaven, but when you put them all together, then you see very clearly the ones that do and how the others are entirely supportive of this belief in dying then going to heaven. Please note that I will make no further comment to anyone on this answer because I am weary of individuals trying to debate points / verses and to justify their question when an answer does not entirely agree with them. You asked for "the biblical basis" and that is what I have provided.