To achieve statehood for DC, would an amendment to the Constitution need to be passed, or would it just work like a regular territory, i.e. only needing Congress to pass a law?
Washington, DC could become a state with just a law, but its representation in the electoral college would not be able to increase properly with population due to the twenty-third amendment:
Section 1. The District constituting the seat of Government of the United States shall appoint in such manner as the Congress may direct:
A number of electors of President and Vice President equal to the whole number of Senators and Representatives in Congress to which the District would be entitled if it were a State, but in no event more than the least populous State; they shall be in addition to those appointed by the States, but they shall be considered, for the purposes of the election of President and Vice President, to be electors appointed by a State; and they shall meet in the District and perform such duties as provided by the twelfth article of amendment.
So long as Washington, DC remains the seat of government of the US, it would be limited to the number of electors in the least populous state. Unless of course this amendment were considered moot in the situation of statehood. Presumably most of it would be rendered moot (otherwise the electoral college would get rather goofy with the DC electors being both in addition and part of those appointed by the states). Perhaps that clause would as well.
It would be more reliable to repeal the twenty-third amendment in that circumstance.
Aside from the 23rd Amendment issues that have been mentioned in other answers, there are some additional parts of the Constitution that would cause me to argue that an amendment to the Constitution would indeed be required in order for D.C. to become a state (at least as long as it remained the seat of government.)
I would argue that Article I, Section 8 prevents the seat of government from being in any state. It says,
The Congress shall have the power...
To exercise exclusive legislation in all cases whatsoever, over such District (not exceeding ten miles square) as may, by cession of particular states, and the acceptance of Congress, become the seat of the government of the United States, and to exercise like authority over all places purchased by the consent of the legislature of the state in which the same shall be, for the erection of forts, magazines, arsenals, dockyards, and other needful buildings;--And
To make all laws which shall be necessary and proper for carrying into execution the foregoing powers, and all other powers vested by this Constitution in the government of the United States, or in any department or officer thereof.
It seems pretty clear (to me, at least) that the intent here is that the seat of the government of the United States is to not be part of any state and that Congress shall have exclusive legislative control over it, unlike with the states.
Additionally, Article IV, Section 4 says,
The United States shall guarantee to every state in this union a republican form of government, and shall protect each of them against invasion; and on application of the legislature, or of the executive (when the legislature cannot be convened) against domestic violence.
The guarantee that every state would have a republican form of government necessitates that the state would have its own legislature. If D.C. were to become a state while remaining the seat of government, it could not simultaneously satisfy both this requirement and the above-listed requirement of Article 1, Section 8 that Congress have exclusive legislative control over the seat of government. Therefore, it would not be constitutional for D.C. to become a state while remaining the seat of government.
Of course, if Congress decided to move the government somewhere else - and one or more states agreed to cede that territory to the federal government for the creation of a new seat of government district - then D.C. could be admitted as a state without a further amendment to the Constitution once it was no longer the seat of government. Alternatively, Congress could choose to give the remainder of the District of Columbia back to Maryland after the seat of government was moved, just as in 1847 Congress gave back the portion of the District that had been ceded by Virginia back to Virginia.
Yet another possibility would be for most of the remaining part of D.C. to be returned to Maryland, while leaving only the federal lands around where the White House, Capitol, and Supreme Court reside in a federal seat of government district. This could be done without an amendment and has actually been proposed quite a few times over the years, including relatively recently. The only part of this that might require an amendment would be repealing the 23rd Amendment, so that the handful of people (the President and maybe a few others?) who still lived in the federal portion wouldn't have 3 electoral votes to themselves.