To say when Americans in general started to think that Civil War is inevitable it's hard. There were some who believed that the Union was doomed since the great debacle of 1848 that ended with the Compromise of 1850. There were others who believed that another compromise could still avert Civil War until shortly before Fort Sumter fell. But, I'd say that John Brown's raid on Harpers Ferry was the turning point, that was when many moderates turned away from the Union and started to expect secession. It was at that point that the South concluded that they could never submit to Republican rule, and with the Democratic Party bitterly divided, the election of Abraham Lincoln was all but assured.

The John Brown raid caused so much panic in the South because it was a strike against their sensibilities. Southerners lived in constant fear of slave insurrections. It's paradoxical, really, because Southern propaganda always insisted that slaves were happy and better off than Northern laborers. Why, then, would they revolt against a system that protected them like that? Southerners concialited the absurd belief that slaves were happy with the possibility of slave revolts by saying that the later were caused by Northern agitators, who in pursuit of fanatical abolitionism would liberate the slaves and set them at their throats. Moderates scoffed at this belief. Even if they were angry at the North for not respecting their "rights" (read: expanding slavery, and violating the rights of Northern states), there were many Southerners who came from a Whig background of compromise and moderation, who still believed that Northerners prized the Union and respected the Constitution enough for them not to worry about slavery in the Slave States.

Then John Brown struck. It's not really worth it to go in detail over the raid. The important fact is that John Brown was aided by a group of Northerners with the absurdly cool name of the "Secret Six", and that his plan was not merely to liberate a few slaves, but to start a gigantic rebellion that would destroy the South and its society once and for all. John Brown failed, and he was hanged for the crime of treason and inciting insurrection (they hung him for a traitor, they themselves the traitrous crew!). But his ideals and his message resonated with Northerners. You see, the South believed that the Republican Party, dedicated to preventing the spread of slavery, was a direct threat towards their society, liberty, and safety. But Northerners too believed that the South was an existential threat. They believed that the Slave Power, a shady conspiracy to spread and protect slavery, would force them to accept the hated institution, thus destroying the republican liberty and free enterprise they cherished as the foundations for the best society in God's footstool.

Northerners were tired of being impotent against the Slave Power. The Slave Power had trampled their rights and send slave catchers North to kidnap their neighbors. It had hijacked the Supreme Court to issue a ruling that completely went against their ideology, perpetuating slavery. It had almost forced slavery down in Kansas. And time again and again, they were powerless. But here came John Brown, a fearless man who struck a direct hit against this Slave Power.

Northerners were at first confused by this, and they condemned it as a mad enterprise. But as the trial continued, Brown successfully portrayed himself as a selfless savior who struck a blow for freedom. His final words were that if it was necessary to "forfeit my life for the furtherance of the ends of justice, and mingle my blood further with the blood of my children and with the blood of millions in this slave country whose rights are disregarded by wicked, cruel, and unjust enactments, I say, let it be done​." These words caused a profound sentimental reaction within the Northern heart. And fury within the Southern heart, as well.

The North "has sanctioned and applauded theft, murder, treason," declared the popular DeBow's review of Louisiana. A Richmond newspaper declared that "the Harpers Ferry invasion has advanced the cause of disunion more than any other event,’’ while a Baltimore newspaper said that the South could no longer "live under a government, the majority of whose subjects or citizens regard John Brown as a martyr and a Christian hero." Henry Wise said that they could not suffer any other such raid “without suffering worse than the death of citizens—without suffering dishonor, the death of a State.” An Alabamian, who had considered himself an “obstinate Unionist under the impression that Republicans are not abolitionists,” said that he now felt “smothered by the want of sympathy from the North. . . . The time has come at last for war, or the Yankeys [sic] must let us alone.” A Virginia woman concluded sadly that ‘‘there is no denying that our people were in a palpable state of war from the time of the John Brown raid.’’ A newspaper made the observation that the raid "wrought almost a complete revolution in the sentiments . . . of the oldest and steadiest conservatives. . . . Thousands of men . . . who, a month ago, scoffed at the idea of a dissolution of the Union . . . now hold the opinion that its days are numbered."

Indeed, the John Brown unleashed a wave of resentment, fear and bitterness hitherto unseen. McPherson even compares it with the Great Fear that gripped the French countryside during the Revolution, when the people believed that the King's brigands were coming. Well, the South believed that the Black Republican legions were coming, and they answered by forming militias, expelling Yankees violently, and attacking the North with vehement hatred. They said that the raid was the natural result of Northern fanaticism, and vowed to resist to the end. They would "never permit this Federal government to pass into the traitorous hands of the Black Republican party," thundered Robert Toombs. ‘‘If, in our present position of power and unitedness, we have the raid of John Brown . . . what will be the measures of insurrection and incendiarism, which must follow our notorious and abject prostration to Abolition rule at Washington[?],’’ asked a newspaper. Another declared that anybody "who does not boldly declare that he believes African slavery to be a social, moral, and political blessing" was an enemy of the South.

This climate of "fear and hostility" caused many Southerners to believe that the Union was doomed. Congressmen came armed to the sessions, and fights often broke out. Even a pacifist clergyman bought a revolver to defend himself. A Southerner declared that he couldn't "help wishing the Union were dissolved and we had a Southern confederacy", while another thought it better that "the wheels of government should stop [and the Union] demonstrate itself to be a failure and find an end, than our principles, our honor be infringed upon." The Governor of South Carolina threatened to send militias to Washington. It was at this point that many Southerners believed that secession was inevitable. Vowing to never submit to the Yankee "blackguards", Southerners were ready to secede as soon as a Republican was elected president. And since the Democratic Party was bitterly divided, and someone could win without carrying a single Southern state, the election of Lincoln was almost secured.

Nonetheless, Northerners did not believe that Civil War was coming. They thought that Southern threat were bluffs, used to push they around as they had done so many times, that it was "the old game of scaring and bullying the North into submission to Southern demands," They were sure that at the end Southern moderates would win, and the Union would survive. Events would prove them wrong, but this belief that a secret Unionist majority existed in the South would remain during the first two years of the war. The votes for Douglas can be seen as people who believed that a Lincoln victory would cause civil war. But it seems that a Northern majority did not believe secession to be imminent.

When Lincoln won, the South seceded. Unionionist in the Border States and Washington tried many combinations to try and save the Union, still clinging to the belief that war could be averted. The only promising possibility was a compromise that explicitly protected slavery, but Republicans refused to follow through because they saw it as a repudiation of democracy and an unconditional surrender. It was at this point that Northerners were finally convinced that Civil War was indeed inevitable, and they started to prepare for it.

To summarize, Southerners became convinced that secession (and perhaps Civil War) was inevitable after John Brown's raid, which caused such hostility and fear within Southern minds that any Republican victory would justify secession. Northerners and some Unionists at first scoffed at this idea, but after the South truly seceded and the only compromise possible was a surrender, they also became convinced that a Civil War was inevitable, because they could never surrender the government.

Sources:

The Battle Cry of Freedom by James McPherson.

Disunion! The coming of the Civil War by Elizabeth R. Varon.

The Road to Disunion by William Freehling.