By conditionalist doctrine, I mean the belief that hell is the destruction of those souls that go there, rather than the everlasting torment or eventual purification of said souls.
Technically, a couple of different terms are being confused in the question.
Conditionalism is the teaching that immortality / eternal life is conditional upon a right standing with God.
Annihilationism is the teaching that anyone who is ultimately unrepentant will cease to exist, usually after a period of torment that accommodates to the amount of evil they have done.
While these are technically separate, they do often go hand-in-hand.
Most defenses for conditionalism start out with a general summary of their main point: nothing in the bible explicitly says humans are inherently immortal; not their bodies, not their spirits, not their souls.
Specific appeal is made to a selection of texts that seem to illustrate as much. Some key texts include:
What Is Death?
Because conditionalism and annihilationism are teachings involving 'death', the discussion of course requires defining what 'death' consists of. It is frequently said by advocates of the traditional view ('eternal conscious torment') that 'death' should be defined as 'separation'. Bodily death is separation of the spirit/soul from the body, and spiritual death is separation of the spirit/soul from God. In both cases, the spirit/soul is believed to continue existing consciously in its state of death.
Conditionalists generally criticize this as a redefinition of the word. 'Death', they say, has always been defined as a cessation of life, consisting of a lack of all consciousness, comparable to a dreamless sleep. To support this definition, conditionalists again point to a selection of biblical texts:
This is in addition to the numerous texts throughout the bible that compare death to 'sleep', no matter who it is being described (e.g. 1 Kings 2.10; First Kings 22.40; John 11.11-14; First Thessalonians 4.13f).
With this altogether, conditionalists believe the bible broadly teaches that humans are not innately immortal, in any way, and that their natural end is to die and cease to exist.
Where conditionalism defines human immortality as conditional upon a right relationship with God, annihilationism is defined as a direct punishment of death from God. Qualitatively, there is no distinction between 'death' and 'annihilation'; the latter word is used solely to clarify just what it is that 'death' consists of.
Again, on a broader level, annihilationists believe the bible teaches that humans who are ultimately unrepentant will suffer death / cessation of existence. Poetic idioms in the Psalms are said to accurately describe a lack of existence, prophetic metaphors are said to capture the essence of a lack of existence, and the 'plain meaning' of basic words are said to describe a lack of existence directly.
The final fate of the unsaved is:
It is widely agreed, even by critics, that the best, complete treatment in favor of conditionalism and annihilationism is The Fire That Consumes, written by Edward Fudge. While it is difficult to speak on behalf of a broad spectrum of belief, Fudge's book is yet considered the masterwork defense of these two beliefs.
This is going to be an incomplete answer because this question is broad and is connected to other doctrines like the nature of man, the state of the dead and the judgment.
Since the beginning of sin there was only two possibilities for mankind: life as offered by Christ or death. It was not eternal life in heaven or eternal life in hell.
Genesis 2:17 But of the tree of the knowledge of good and evil, thou shalt not eat of it: for in the day that thou eatest thereof thou shalt surely die.
Romans 6:23 For the wages of sin is death; but the gift of God is eternal life through Jesus Christ our Lord.
John 3:16 For God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish, but have everlasting life.
These verses above are referring to the second death and not the first death.
John 8:51 Verily, verily, I say unto you, If a man keep my saying, he shall never see death.
The second death is complete destruction of the wicked in the lake of fire as described in the verses below.
Psalm 37:20 But the wicked shall perish, and the enemies of the Lord shall be as the fat of lambs: they shall consume; into smoke shall they consume away.
Matthew 10:28 And fear not them which kill the body, but are not able to kill the soul: but rather fear him which is able to destroy both soul and body in hell (γέεννα).
2 Thessalonians 1:8,9 In flaming fire taking vengeance on them that know not God, and that obey not the gospel of our Lord Jesus Christ: Who shall be punished with everlasting destruction from the presence of the Lord, and from the glory of his power;
Malachi 4:1,3 For, behold, the day cometh, that shall burn as an oven; and all the proud, yea, and all that do wickedly, shall be stubble: and the day that cometh shall burn them up, saith the Lord of hosts, that it shall leave them neither root nor branch. And ye shall tread down the wicked; for they shall be ashes under the soles of your feet in the day that I shall do this, saith the Lord of hosts.
Sodom and Gomorrha are set forth as examples of suffering eternal fire in Jude, but they were clearly destroyed.
Jude 1:7 Even as Sodom and Gomorrha, and the cities about them in like manner, giving themselves over to fornication, and going after strange flesh, are set forth for an example, suffering the vengeance of eternal fire.
2 Peter 2:5,6 And spared not the old world, but saved Noah the eighth person, a preacher of righteousness, bringing in the flood upon the world of the ungodly; And turning the cities of Sodom and Gomorrha into ashes condemned them with an overthrow, making them an ensample unto those that after should live ungodly;
We are judged by our works so this means that everyone will have different rewards not the same reward. Revelations 20:13,14 And the sea gave up the dead which were in it; and death and hell delivered up the dead which were in them: and they were judged every man according to their works. And death and hell were cast into the lake of fire. This is the second death.
The word forever and eternal (olam in the old testament and aionios in the new testament) in the bible does not necessarily mean an endless length of time it can also mean an age. Take for example Jonah 2:6 and 1 Samuel 1:22. In both these cases forever is clearly referring to a specific amount of time 3 days and nights in Jonah and to the end of Samuel’s life in Samuel. It really depends on the context of the verse itself. In many cases can just when dealing with the punishment of the wicked it could mean an ambiguous time until all the wicked are burned up or it could be referring to the nature of the punishment which has eternal consequences (eternal separation from God in death).