If a U.S. citizen who has another citizenship enters Cuba using that other citizenship and spends his vacation there, will he run afoul of U.S. law (embargo of Cuba)?
The only people the law does not apply to are " persons licensed by the US Treasury Department’s Office of Foreign Assets Control." http://www.visitcuba.com/travel-guide/travel-tips/special-note-to-usa-travellers/
If you have close family there the license is not needed.
If you qualify to travel on a general license for family travel, you do not need to seek permission from OFAC. However, it is a requirement of the general license that you be able to document, if asked, how you qualify under the general license.
There are also special permissions for certain groups.
There are 12 categories of people who are allowed to visit including close relatives of Cubans, academics, those traveling on official government business, those on humanitarian or religious missions, journalists and people on accredited cultural education programs.
Those rules apply to people with dual citizenship. New rules are now in place, that allow you to visit under certain circumstances considered to not be simple tourism. https://www.forbes.com/sites/alexandratalty/2018/04/23/yes-you-can-still-travel-to-cuba/#34139656378d