Suppose the citizens of one of the states constituting the United States, for whatever reason, wanted to separate from the United States. How could they do that?
So far in U.S. history there is only one precedent. The states of South Carolina, Mississippi, Florida, Alabama, Georgia, Louisiana, Texas, Virginia, Arkansas, Tennessee and North Carolina, by the end of 1861, withdrew from the United States of America by votes within their states. President Abraham Lincoln, supported by a Republican Party-controlled Congress, simply made war (the Civil War) on the untied states, and the federal government won that war.
Since then the predominant thinking in the re-United States has been that a state cannot separate from the Union because military force will be used by the federal government to overcome any such manifestation of local democracy. Kind of like when Czechoslovakia tried to opt out of the Soviet Block. Because the issue of slavery was entangled with the issue of separation of states during the Civil War, it is still hard to have a rational discussion of the topic.
Clearly the U.S. Constitution allows for an amendment that would either allow a particular set of states to separate, or would set rules for separation in general. In fact, one could dissolve the entire Union with a Constitutional amendment. However, before writing such an amendment, we should look at the Constitution as it now stands.
Reading the Constitution, you will not find a single phrase that says that a state cannot withdraw from the union. Nor does it say that a state can withdraw. This is not that different from many issues of federal powers. How broad the specific powers granted the the Federal government are to be in practice has been a constant source of argument and litigation since the ink was dry on the original Constitution.
A majority of Americans opposed the Constitution as written at the time. To get it passed by the nine required states a great deal of bribery and intimidation was used. Also, it was promised that a number of amendments would be appended to the Constitution, to make the package more attractive.
Regarding the separation of states, clearly Amendment X, the Tenth Amendment, has the greatest bearing. Many people voted for the new Constitution with the idea that their state could withdraw from the Constitution if the whole experiment did not work out. They feared a central government that could become as tyrannical as the British Empire had been. Hence the amendment: "The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people."
I don't think you can get more clear than that without illegally declaring war on your neighbors, as Abraham Lincoln did. No where in the Constitution did it say that the Federal government could prevent a state government from withdrawing. The power to decide whether to be part of the United States was clearly a power "reserved to the States."
To force people to be governed against their will by any government, including the Federal government of the United States, is against the natural right of the people to govern themselves. That was recognized by the architects of the American revolution, or at least was prominent in their rhetoric.
If citizens of a state should ever decide they want to separate from the United States, I suggest they go about it with more care than the State of South Carolina did in 1860.
First, the government of the State in question should write a state law saying that it has they right to separate and is indeed separating. Then this should be the basis of trying a case in the Supreme Court of the United States.
The Supreme Court is most likely, in fact certain, to rule that the State cannot secede by virtue of its own legislation. The point is not to expect an honest verdict, but to get a statement of why they cannot rule themselves, and of the construction of the Constitution the court uses to rationalize this un-democratic ruling.
Then, having been ruled to still be part of the United States, the state needs to introduce legislation in the U.S. Congress allowing it to leave. Its leaders should also try to pass a U.S. constitutional amendment making it clear what the rules are for leaving.
Of course, this whole path could be cut off by a Constitutional Amendment saying, roughly, "no State shall be allowed to separate from the United States, and the President and Congress shall have the power to use the military to enforce this provision."
The United States went to a great deal of trouble to break up the former nation of Yugoslavia just a couple of decades ago. We broke Vietnam in half in the 1950's when we did not like the projected outcome of elections there. I don't see why U.S. states should not be granted the same rights of divorce that the U.S. empire has so gladly imposed on weaker nations.
On the other hand, no one should want a messy divorce. Any such separation should be mutual and amicable.
But, given sufficient unaddressed grievances, a Declaration of Independence, backed by the right to self-defense, may be worth a try.