As a matter of fact, we do! Marx's correspondence has been available in English for a while now thanks to the Marx & Engels Collected Works (abbreviated MECW), so in addition to his public writings on Lincoln we have his private thoughts too. But first, some background on Marx's opinion of Lincoln while he was alive.
During the Antebellum, Marx's political contacts in America were mostly of a pro-Republican bent. Many veterans of the 1848 revolution in Germany fled to America, including Marx's friend Joseph Weydemeyer (who later became an officer in the Union army); Marx himself even considered emigrating to America. Many of these former revolutionaries took their democratic and largely anti-slavery politics with them. During the 1850s, Marx eked out a fairly meager living by writing for the pro-Republican New York Daily Tribune, which was founded by Horace Greeley a decade earlier.
With the outbreak of the secession crisis in late 1860 and early 1861, Marx and his friend Friedrich Engels began to pay close attention to the situation in America. Marx's (and Engels') opinion of Lincoln during this period followed from his conduct of the war — and, in particular, the question of slavery. Engels was skeptical of Lincoln's willingness to move against slavery, and hence was more pessimistic about Lincoln, at least early in the war; take, for instance, his letter to Marx in mid-1862:
They shrink from conscription, from resolute fiscal measures, from attacking slavery, from everything that is urgently necessary; everything's left to amble along at will, and, if some factitious measure finally gets through Congress, the honourable Lincoln hedges it about with so many clauses that it's reduced to nothing at all. .... As I have already said, unless the North instantly adopts a revolutionary stance, it will get the terrible thrashing it deserves — and that's what seems to be happening.^1
Marx, by contrast, was more confident that Lincoln would be forced into "revolutionary methods,"^2 and indeed was pleased by the Emancipation Proclamation and other measures in late 1862:
The fury with which the Southerners are greeting Lincoln's acts is proof of the importance of these measures. Lincoln's acts all have the appearance of inflexible, clause-ridden conditions communicated by a lawyer to his opposite number. This does not, however, impair their historical import ...^3
Both men were, in private, critical of Lincoln's perceived timidity — Marx refers to his "lawyer-fashion"^4 — but by 1864 their opinion of Lincoln had improved sufficiently that Marx felt it appropriate to write his famous address to Lincoln on behalf of the International Workingmen's Association congratulating Lincoln on his conduct in the fight against the Confederacy.
The first mention of Lincoln's assassination in Marx's correspondence comes on May 3rd, 1865, where Marx writes to Engels that
The chivalry of the South has ended worthily. In addition, Lincoln's assassination was the most stupid act they could have committed. Johnson is stern, inflexible, and revengeful and as a former poor white has a deadly hatred of the oligarchy. He will make less fuss about these fellows, and because of the treachery, he will find the temper of the North commensurate with his intentions.^5
So, whereas Marx perceived Lincoln to be somewhat milquetoast, he initially perceived Johnson to be a more decisive figure. Publicly, Marx signed the address of the International to President Johnson, proclaiming Lincoln a martyr to the cause of anti-slavery and urging Johnson to take up where he left off.
By June 1865, however, Marx's opinion on Johnson had soured as he realized Johnson was not really going to punish the old Southern enslaver elite:
Johnson's policy likes me not. A ludicrous affectation of severity towards individuals; hitherto excessively vacillating and weak when it comes down to it. The reaction has already set in in America and will soon be much fortified if the present lackadaisical attitude is not ended immediately.^6
By the next year Marx was openly celebrating Johnson's defeat:
You will have been just as delighted by the defeat of President Johnson in the latest elections as I was. The workers in the North have at last fully understood that white labour will never be emancipated so long as black labour is still stigmatised.^7
So, while Marx was initially excited about the prospects of Lincoln's "stern" successor, his hopes were quickly dashed.
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Marx publicly mourned Lincoln's death and hailed him as a hero. A May 1865 letter from the Working Men's International Association to President Johnson that is signed by Marx (and attributed to him) reads, in part:
It is not our part to call words of sorrow and horror, while the heart of two worlds heaves with emotion. Even the sycophants who, year after year, and day by day, stick to their Sisyphus work of morally assassinating Abraham Lincoln, and the great Republic he headed, stand now aghast at this universal outburst of popular feeling, and rival with each other to strew rhetorical flowers on his open grave. They have now at last found out that he was a man, neither to be browbeaten by adversity, nor intoxicated by success, inflexibly pressing on to his great goal, never compromising it by blind haste, slowly maturing his steps, never retracing them, carried away by no surge of popular favour, disheartened by no slackening of the popular pulse, tempering stern acts by the gleams of a kind heart, illuminating scenes dark with passion by the smile of humour, doing his titanic work as humbly and homely as Heaven-born rulers do little things with the grandiloquence of pomp and state; in one word, one of the rare men who succeed in becoming great, without ceasing to be good. Such, indeed, was the modesty of this great and good man, that the world only discovered him a hero after he had fallen a martyr.
The letter even honored Secretary of State Seward as a secondary victim of the assassination attacks, applauding Seward's antislavery rhetoric from the 1850s as well as his more recent service. (Seward was stabbed in the face and probably survived only because of a metal neck brace that he was wearing from a recent carriage accident.)
Marx famously wrote to Lincoln while Lincoln was still alive in the same format, congratulating Lincoln (actually, the whole country) on Lincoln's re-election, and expressing solidarity with the struggle against slavery:
As the American War of Independence initiated a new era of ascendancy for the middle class, so the American Antislavery War will do for the working classes. [We] consider it an earnest of the epoch to come that it fell to the lot of Abraham Lincoln, the single-minded son of the working class, to lead his country through the matchless struggle for the rescue of an enchained race and the reconstruction of a social world.
(Marx would express similar sentiments in the preface to the first German edition of Das Kapital, volume 1.) Lincoln produced a friendly reply through Charles Francis Adams, his ambassador to Britain:
So far as the sentiments expressed by [your letter] are personal, they are accepted by [Lincoln] with a sincere and anxious desire that he may be able to prove himself not unworthy of the confidence which has been recently extended to him by his fellow citizens and by so many of the friends of humanity and progress throughout the world.
For his part, Lincoln sometimes expressed sentiments that seem kind of proto-Marxist. In 1846 or 1847, for example, he wrote in a note:
In the early days of the world, the Almighty said to the first of our race "In the sweat of thy face shalt thou eat bread"; [Genesis 3:19] and since then, if we except the light and the air of heaven, no good thing has been, or can be enjoyed by us, without having first cost labour. And, inasmuch as most good things are produced by labour, it follows that all such things of right belong to those whose labour has produced them. But it has so happened in all ages of the world, that some have laboured, and others have, without labour, enjoyed a large proportion of the fruits. This is wrong, and should not continue. To secure to each labourer the whole product of his labour, or as nearly as possible, is a most worthy object of any good government.
At the time, Lincoln was writing about protective tariffs, but later he would apply the same logic to slavery.
He also included the following passage in his first Annual Message to Congress (the closest thing at the time to the State of the Union address), which sounds strikingly Marxist:
Labor is prior to, and independent of, capital. Capital is only the fruit of labor, and could never have existed if labor had not existed. Labor is the superior of capital, and deserves much the higher consideration.
When Lincoln wrote his earlier note on tariffs, it seems somewhat unlikely that Lincoln would have known who Marx was or been familiar with his ideas. But at the time of the Annual Message, Lincoln probably did know who Marx was and had probably read some of his writing. By that point, Marx was a long-time contributor to Horace Greeley's New-York Daily Tribune, one of the leading Whig/Republican newspapers in the country, which Lincoln is known to have read. Greeley had played a significant role in preventing Seward from getting the Republican presidential nomination in 1860; and Charles Dana, who recruited Marx to the Tribune, went on to work in Lincoln's War Department, rooting out corruption and serving as a liason between General Grant and Stanton (the Secretary of War).
In one way, though, Lincoln conspicuously diverged from Marxist thought. In his address to the Milwaukee Agricultural Society in 1859, Lincoln said:
The prudent, penniless beginner in the world, labors for wages awhile, saves a surplus with which to buy tools or land, for himself; then labors on his own account another while, and at length hires another new beginner to help him. This, say its advocates, is free labor—the just and generous, and prosperous system, which opens the way for all—gives hope to all, and energy, and progress, and improvement of condition to all. If any continue through life in the condition of the hired laborer, it is not the fault of the system, but because of either a dependent nature which prefers it, or improvidence, folly, or singular misfortune.