"A heartbeat away" is a figurative way to say that the VP is the first in line to the Presidency if the sitting President dies. That is, after the President's heart beats for the last time (one heartbeat) and then dies, the VP then becomes President.

I agree with both Brian and Kiara, but I want to add that the author is cleverly playing with words in this paragraph and both the literal and figurative meaning are being used. There is the saying "a heartbeat away" which means very close or near (you can get there or it can happen in the time it takes for a heart to beat a single time). However, "heartbeat away" is also being used literally here. When the President's heartbeat stops (he dies), then the Vice President will become President. So it's very clever to use the same words to describe the situation both figuratively and literally. Just one last note on the word "heartbeat." It can be used to describe each individual beat of a heart, or it can be used to describe the rhythm of your heart (the constant beating and its pattern). I might say "My heartbeat is fast" (referring to overall rhythm. Or I could say, "I can hear each heartbeat" (referring to the individual beats).

Something like : It's very close.. it's near