This is an interesting question. The most obvious way that the two characters are alike is that both of them get put into harsh conditions--against their will, and most likely caused by human weakness, cruelty and war--in which they have to survive. They did not choose to be in those situations, and in their environments, they have to survive against nature and human brutality. Katniss must fight for her life as others try to kill her, and the father has to do the same, with the added stress of caring for his son along the way. It is a kill or be killed environment that they have to survive in.
Another similarity between these two characters is their fortitude in difficult situations. The father, probably not a brutal or aggressive man by nature, hauls a gun around and uses it and other weapons when he needs to in order to survive. Katniss is the same; it's not like she goes around hunting people for fun on her own time, but when required to, she steps up to the plate. They both demonstrate the ability to survive, using fortitude and strength of character.
They are also both protecting people they care about. The father protects his son, and Katniss protects Rue, and eventually Peeta. This drives them to survive even more; they aren't just doing it for themselves, they are doing it for someone else.
Lastly, they both manage to hold on to their humanity throughout their brutal circumstances. The father could have easily turned to barbarism to provide more meat and protection for himself and his son, but he refused, and even taught his son about "the light" that he should carry within him at all times (humanity). Katniss manages to bring humor, love, compassion and selflessness to the games, something that is rarely seen. Despite her circumstances, she remains human, exemplifying the positive traits rather than the negative.
I hope that those thoughts help a bit; good luck!