How does one have standing before the US Supreme Court?

First, the Supreme Court is limited to hearing cases that satisfy the "case or controversy" clause in Article III of the Constitution. Under this "Constitutional Standing" doctrine:

(1) The aggrieved party must have an "injury in fact," which is a particularized or individualized concrete harm that is actual or imminent, (2) The injury must be redressible (the Court can actually do something about that injury), and (3) The injury must be "fairly traceable" to the Defendants' actions (causation).

Beyond this, there is something called "prudential standing," where the court only hears certain types of cases out of sensitivity to its own role. This means that: (1) there is generally no "third party standing" (the court generally restricts standing to those directly injured), and (2) there is generally no standing for "generalized grievances" like taxpayer suits.

Also keep in mind that the Court will not hear cases that are not "ripe" (where it is premature for review), or cases that are "moot" (an actual controversy must be extant in all stages of review). There are exceptions to this mootness doctrine (controversies that are capable of repetition, yet constantly evading review, like abortion), and Congress can make exceptions to these prudential standing doctrines (by making a derivative cause of action, for example).

Other than the issue of "standing" in general, a party brings a case before the U.S. Supreme Court usually by going through the Trial Court, appealing to the Intermediate Court, and then seeking a writ of certiori. If the Supreme Court GRANTS certiori, then the case will be heard by the Court. In a few very rare circumstances, the Court has original jurisdiction, but this is discretionary and usually not an issue.

For cases that have shaped the doctrine of standing, see: Lujan v. Defenders of Wildlife, Mass. v. EPA, Allen v. Wright, and FEC v. Akins. Also see Article III of the Constitution.