If the US seceded then why did Lincoln say the CSA was illegal? Wouldn't people have thought this was a bit strange and hypocritical?

The U.S. didn't secede. The U.S. revolted to gain its independence. And the difference is important, and explains why Lincoln wasn't hypocritical. Confederates tried to argue that they weren't revolting, that they weren't engaging in a revolution, but instead, were exercising a U.S. Constitutional right.

Unionists (and the courts in general) had said, and would say again, that there was no Constitutional right to secession. What Unionists did recognize, though, was that the Confederacy was trying to revolt. And there is an extra-Constitutional (i.e., illegal) "natural right" of revolution, just as happened during the American Revolution.

Lincoln made this distinction very clear in his First Inaugural Address, stating:

"This country, with its institutions, belongs to the people who inhabit it. Whenever they shall grow weary of the existing Government, they can exercise their constitutional right of amending it or their revolutionary right to dismember or overthrow it."

But exercising the right of revolution does not come without legal repercussions. Revolution is simply the illegal overthrow of government through force of arms. But as the U.S. Constitution makes clear, one of the primary responsibilities of the federal government is to uphold the U.S. Constitutional rule of law.

Therefore, if the Confederacy would not exercise their constitutional rights to change the government (e.g., lawsuits in courts of law, future elections that would throw the current office-holders out of office, Constitutional Amendments or other legislation that would change the laws, etc.) and instead were to try to engage in an unconstitutional "natural right" to change the law through force of arms, then it was the President's Constitutional duty to uphold the Constitution by any and all means necessary. As Lincoln said in his same speech, the President takes an oath of office to "preserve, protect, and defend" the Constitution, and he did just that.

The previous president, James Buchanan, had essentially said the same thing a few months earlier in his final State of the Union ("Message To Congress") Address:

"I have purposely confined my remarks to revolutionary resistance, because it has been claimed within the last few years that any State, whenever this shall be its sovereign will and pleasure, may secede from the Union in accordance with the Constitution and without any violation of the constitutional rights of the other members of the Confederacy; that as each became parties to the Union by the vote of its own people assembled in convention, so any one of them may retire from the Union in a similar manner by the vote of such a convention....

"Such a principle is wholly inconsistent with the history as well as the character of the Federal Constitution....

"Secession is neither more nor less than revolution. It may or it may not be a justifiable revolution, but still it is revolution."

He actually goes on at some length on the topic, and it's worth reading in full. It largely echoed the legal arguments already made against secession and nullification by Andrew Jackson thirty years earlier in his Nullification Proclamation.

In other words, nobody ever denied the South had a "natural right" to engage in revolution. But as James Madison had argued against the constitutionality of secession (see his Notes on Nullification from 1830, 1834, and 1835), he said that any revolutionary option invites the force of arms in response. And if the revolutionaries lose in this battlefield contest, then they will still be under the rule of law they attempted to unconstitutionally overthrow, only now they've broken a bunch of laws that will likely lead to trials for treason, death sentences, and other legal repercussions.

So, no, there was no real hypocritical point of view coming from Abraham Lincoln or other Unionists. They very much made the distinction between a Constitutional right, and a revolutionary one, with the latter one subject to be put down by the several provisions in the Constitution. The provisions empower the President and Congress to suppress insurrections and rebellions, to ensure that every state has a "republican form of government" (such as respecting the electoral process after a legally-binding Presidential election), and commands that President "preserve, protect and defend the Constitution of the United States".