In the United States, which laws would spying on someone using their computer's webcam, without their permission and in private areas, break?
A creative prosecutor could probably come up with a raft of charges. But you could start with the federal wiretapping statute, 18 USC 2511, and the anti-hacking statute, 18 USC 1030. Here is an indictment brought in 2012 under the anti-hacking statute against someone who distributed and used this kind of software.
Depending on the facts and the jurisdiction, this may also constitute the tort of intrusion on privacy or seclusion, a tort recognized by the Restatement (Second) and actionable in many jurisdictions. The most common test is whether the invasion would be "offensive to a reasonable person."
And no, contrary to the commenter's view, a "click to accept" license is not a get out of (literal) jail free card here. Courts interpret adhesion contracts liberally to favor the signer, and outrageous terms hidden in small print are not guaranteed to be enforceable, especially if the software is clearly designed to trick people into installing it. The license terms might even hurt you, by providing evidence of your intent to use the software for perving rather than its ostensible use.
This is not an exhaustive list, and there may be additional state-level statutes that apply. Bottom line: yes, this is clearly illegal, and the courts will be reluctant to let you trick people into getting away with it.