Reactions varied depending on who you asked. Political leaders and Confederate generals (i.e., those with a more firm grasp on the realities of the assassination's implications) all seemed to understand that the effects on the Confederacy (which was a month away from non-existence) would be detrimental, not beneficial. Lincoln was shot on April 14th, and news would have started getting out about it in the Confederacy as early as the next day in some places. Although Johnston's army was still in the field, Lee had surrendered on the 9th, and Richmond had fallen, so there wasn't any realistic hope that Lincoln's death would provide a boon to southern fortunes in terms of prolonging the war.

The only hope the Confederacy had in winning its independence after 1863 was via a surge in the northern peace movement (led by the Copperheads). This played on northern war fatigue, and a desire to stop the bloodshed. If anything, Lincoln's murder revived northern bloodlust, and re-inspired the war-weary north to exact its vengeance and will on a mostly-defeated enemy that might have otherwise enjoyed more lenient, amicable terms. Understand, with Booth killed, and the conspirators hanged (except for John Surratt), the north had only the south to punish for Lincoln's death, and this they did through the installment of military governors and aggressive legislation (well-intentioned though it may have been) to confront racial prejudice in the south.

There was scarcely an educated man or woman that didn't understand that the south as a whole would suffer for Lincoln's assassination. The average layperson in the south might have rejoiced at the news of the assassination, and there are some stories about people throwing parties to celebrate, but I don't think that the evidence supports this as representative of the majority. Those who had been touched by war, and understood the realities of the politics at play, understood that Lincoln's death meant more deprivations for the southern people. Jefferson Davis didn't seem to mourn Lincoln all that much, although R.E. Lee made a few remarks making his disapproval of the act clear. Some southern primary sources from average folk (diaries, newspapers, etc) do express a sentiment along the lines of "he got what he deserved," but again, by and large, it appears that most people understood that it was, on the whole, a bad development.

[Source: Jay Winik, 'April, 1865']