Finding five is not easy. Outside of Lincoln and FDR, you're basically talking about either placeholder presidents or those who did lots of damage.
There were some accomplished men whose best achievements were outside the presidency.
Jefferson
Eisenhower
Carter
I'd suggest a very cynical standard for both best and worst pres. Body counts, who caused the greatest number of lives to be lost or saved. That includes the betterment or worsening of people's lives through economic improvement or human rights advances/losses.
Thus the choice of Lincoln, who brought the greatest human rights advancement in US history, bar none, the ending of the tyranny and genocide that was slavery.
Thus the choice of FDR, whose New Deal improved lives through more workers rights and lengthening the lives of seniors through social security.
AH choices, based on my criteria:
Henry Wallace for heading off the Cold War.
Eugene Debs for making democratic socialism a reality beyond the half measures of FDR's New Deal.
Eugene McCarthy for ending the Vietnam War and possibly bringing an earlier end to the Cold War.