This is a long standing trope among some Confederate sympathizers and other people who are trying to be as fair as possible. How much of it is actually true?
A fair bit of it is actually true, he suspended the freedoms of speech and press, suspended habeas corpus (arguable legality, the constitution says the government can suspend it during times of rebellion or invasion but it doesn't specify if congress or the president has that power. There are two arguments here, the argument put forth by those opposed to Lincoln's suspension was that the section of the Constitution that this provision is contained in refers to Congress. The other arguement (put forth by Lincoln's supporters) is that the constitution gives the right for habeas corpus to be suspended in cases of emergency and because Congress moves very slowly the President can suspend habeas corpus to solve the crisis quickly) and even imposed martial law incrementally and eventually (in 1863) imposed martial law nationwide. He called up about 75k militia and ordered a blockade of Southern ports. This violates the constitution because this is an act of war, something that only congress can do. Then he called for about 40k volunteers to serve for three years, even though Congress has the right to raise armies not the president under the constitution. Even the Empancipation Proclamation can be argued to be of dubious constitutionality. However that one has considerable leeway because of the concept of the nation being at war, and the fact that the constitution goes out of its way to avoid mentioning slavery. He always maintained that these were only wartime measures and when the war was over he would get rid of all of them, and to be fair, he and his successor Andrew Johnson did away with most of them when the war was done.
Sources:
This is a short summation of "Lincoln's Suspension of the Writ of Habeas Corpus: An Historical and Constitutional Analysis" by James Dueholm
There's no question that Lincoln suspended habeus corpus. Initially it was a localized suspension for the protection of the public. When a Maryland man was arrested by the Union military, Supreme Court Chief Justice Roger Brooke Taney, who happened to be sitting as a trial judge in the area, issued a writ calling for the arrested man to be brought to the court, which the military refused. Taney ruled (in his position as trial judge) that the ability to suspend habeus corpus was limited to Congress because it's in Article I of the Constitution not Article II. Higher level courts never ruled on the issue. Congress did, much later, pass legislation that said that the president had the power to suspend of habeas corpus. However, prior to that Lincoln declared nationwide martial law and suspension of habeus corpus albeit only with regard to members and supporters of the rebellion.
In that there are no court decisions on the subject other than the Taney trial ruling, there's not really any actual judicial opinion on the matter. The academic discussion of the issue tends to favor Lincoln, however.
Separately, there are questions over the constitutionality of emancipation but I've not read much about that.
There are a couple of talking points that can be made regarding Abraham Lincoln and the Constitution during the Civil War. The two that I know most about are his suspension of Habeas Corpus and then his issuance of the Emancipation Proclamation.
On the first issue, people will often discuss that Lincoln unlawfully suspended the writ of habeas corpus during the war thus violating American Constitutional rights. Habeas Corpus protects citizens from unlawful imprisonment, imprisonment without trial, and imprisonment without sentencing. Shortly after war breaks out in April 1861, Lincoln and other "Union officials . . . continue to worry about underground Confederate activities in Baltimore" (Battlecry of Freedom, McPherson, p.289, 1988). As a result, several people suspected of supporting disunion in Baltimore are arrested, and appeal to the circuit courts citing the writ of habeas corpus but aren't successful because Lincoln suspended it on April 27th.
Lincoln is challenged by the Supreme Court Chief Justice Taney as well as Congress over this suspension, but getting back to the idea of Constitutionality will help us understand it from the perspective of your question. Article 1, section 9, clause 2 of the United States Constitution states: "The privilege of the Writ of Habeas Corpus shall not be suspended, unless when in Cases of Rebellion or Invasion the public safety may require it." With the secession of southern states, the forming of the Confederate States of America, and the battle at Fort Sumter on April 12-14, Lincoln believed that rebellion and invasion was indeed a threat. This is particularly true along the border states, and in this case, Baltimore, Maryland. So, while there did continue to be challenges to the suspension, from a strict reading of the Constitution and Lincoln's perceived threats he was acting within the law. Two years later, congress would also pass the Habeas Corpus Suspension Act thus giving congressional approval to its suspension during the war.
When thinking about the Constitutionality of the Emancipation Proclamation there are some other ideas to consider. Lincoln knew that during peacetime there was no way he could touch slavery because of the power given to states by the Constitution, but he also never had any official intention to do more than restrict the expansion of slavery.
The war gave Abraham Lincoln the means to address slavery in a way he couldn't prior to the war. In fact, up until the issuance of the proclamation, Lincoln's official stance stated the Civil War was to preserve the Union. However, he wrote within the Proclamation that freeing the slaves was "a fit and necessary war measure" by allowing "that such persons of suitable condition, will be received into the armed service of the United States." By framing it in this way, Lincoln justifies his actions through Article 2, section 2 of the Constitution which gives him title as "Commander in Chief of the Army and Navy of the United States."
To stay within what Lincoln and others perceived as Constitutional Limits, the Emancipation Proclamation only freed slaves in rebelling states. In the Proclamation Lincoln wrote that only in the states "in rebellion against the United States . . . [and who were not] in good faith, represented in the Congress of the United States by members chosen" would slaves be emancipated. He could not Constitutionally change the status of slaves within the states who were not rebelling and did have representation in Congress. Thus, slaves were freed in the Confederate States of America, but not in Missouri, Kentucky, Maryland and Delaware who were still apart of the Union.
The United States Constitution gives rights to the president during wartime when the country is under threat. Lincoln recognized those threats and acted in what he believed to be the best interests of the Union. So, from a strict Constitutional perspective I do not believe Lincoln violated either the letter or the spirt of the Constitution.
What did happen, however, was the shift in the meaning of the war. In the north, officially the war prior to 1863 was about preserving the Union; in the south, it was about states' rights. The Emancipation Proclamation forced people from the North and the South to consider the future of slavery and established a foundation for what would later become the 13th Amendment to the Constitution. From a southern perspective, it is not necessarily unreasonable to conflate these these two together. Especially when you consider that Lincoln wrote in the Emancipation Proclamation that slaves in the rebelling states would be "thenceforward and forever free" and that Lincoln believed freeing the slaves "was an act of justice, warranted by the Constitution." If it was unconstitutional prior to the war to free the slaves, from a southern perspective it would be unconstitutional during the war. But, if you are Lincoln, you are extending Constitutional rights to an hitherto group of people disenfranchised from political and social life in the United States and extending the protections to that group offered by the Constitution.