How do we amend the Constitution?

Well, it's a difficult process, and that's kind of the point; when an amendment is added, the fundamentals of American law shift, in a way that no branch can ignore. The consequences might be relatively minor, as with the 27th amendment prohibiting Congress from raising its own pay until after the next election, or they might be huge, as with the 19th amendment giving women the right to vote in all elections.

The Congress, whenever two thirds of both houses shall deem it necessary, shall propose amendments to this Constitution, or, on the application of the legislatures of two thirds of the several states, shall call a convention for proposing amendments, which, in either case, shall be valid to all intents and purposes, as part of this Constitution, when ratified by the legislatures of three fourths of the several states, or by conventions in three fourths thereof, as the one or the other mode of ratification may be proposed by the Congress; provided that no amendment which may be made prior to the year one thousand eight hundred and eight shall in any manner affect the first and fourth clauses in the ninth section of the first article; and that no state, without its consent, shall be deprived of its equal suffrage in the Senate.

Either way, the basic process isn't too complicated: both the Congress and the states must agree decisively that the amendment is needed. There are two ways to achieve each objective, making for four paths total, but all but one of our amendments gone through the same one.

First, we need the approval of Congress; this can come either from a 2/3 vote in favor in both houses, or a 2/3 vote in favor from a convention called by Congress at the request of the legislatures of 2/3 of the states. The former is straightforward, while the latter is a little more vague. Who would be at this convention, and how many people would be there be? There is no answer, because it has never happened, and Congress has never passed legislation addressing the question either.

Second, we need approval of the states, which can be simple majority votes, but those votes have to take place in 3/4 of the states, either in the legislature itself, or in a state ratifying convention. The same question applies for the latter, and while a few states have enacted laws providing a general method for the conduction of those conventions, for the most part it's up to each state to decide how to do so, if Congress does it again. Congress only specified conventions once, for the 21st amendment, fearing that the criminals making enormous amounts of money from liquor's illegality would pressure state legislators to keep it that way, but conventions would be less susceptible to that.

All well and good, that's how it works. How can I make that happen? you might be thinking, assuming there's an issue dear to your heart that would require a constitutional amendment to correct. The single most crucial thing here is organization; no one ever got an amendment alone. It's quite possible that there's already a group dedicated to your cause, and whether or not there is, you'll have to work very hard to persuade Congress and your state to take it up.

Say you're not up for quitting your job to work on this, or making it your second job. How can you do anything?

• Well, you can contact your representatives, but to make that remotely effective you'll have to get a whole lot of other people to do the same -- Twitter and Facebook help a lot here.
•  This is primary season, so if your state hasn't had them yet, you can vote for people who support your stance on this issue; the most important part here is telling the incumbent "You lost/gained my vote because you don't/do support X!" And yes, this requires you to vote, since politicians have no reason to care about anything you say if you're not part of the political process. 
• Find someone in the legislature who agrees with you, even if they don't represent you, and cheer that person on. 
• Do as much research into the topic as you can, so you can make an informed case for it, and address counterarguments.
• Officially, counties and cities have no role in constitutional amendments, but that doesn't mean they can't influence the legislature. Talk to your board of supervisors or whatever your local government is, and ask them to approve a resolution calling for your amendment.
• If you live close to your state capital, you might be able to attend hearings on legislation relating to your issue; if you also have a dogged crusader for your issue and intimate knowledge thereof, it's just faintly possible that you might be able to testify on the subject.