Above all else Frank wants to achieve a legacy. While most people create a legacy though having children Frank and Clair made the decision to never have children and instead focus on attaining power to create their legacy.
Frank identifies that the main two ways to attain a legacy (without children) are through attaining large amounts of money, or though attaining political power and making major policy changes. Frank identifies that monetary power fades after a generation or so, while major policy changes typically have repercussions for centuries.
When it comes to political power in the United States, it is easily argued that the President is the most powerful person in the nation. And while the current laws say a president can at most serve for 8 years, their polices typically have effects that last for generations. Presidents are also known to hold political influence long after they have left office.
For example Abraham Lincoln was behind the emancipation proclamation which went into effect Jan 1st 1863, but has effected every american generation since, as well as make Abraham Lincoln a household name. Around the same time the richest American was John D. Rockefeller; outside of having a square in New York City named after him, not many american's can easily identify who he was, and his empire has long since faded.
Frank hopes to be the next great president that creates policies that effect generations to come.