Right off the bat, I think you can make a solid case for Katniss in Slytherin or Gryffindor - but I come down on the Slytherin side.
Katniss is like a Slytherin in that she protects her family; she’s hugely focused on her family and the people she knows she loves, and they come first in her life: above everyone else. Katniss is shoved into the role of being a symbol for the rebellion, but it’s not something she wants: she doesn’t want to be the hero or the face of the rebellion, but doesn’t have a choice in the matter.
Katniss is also very Slytherin-ish in that she’ll “use any means to achieve her ends” - she’s cunning, resourceful, and doesn’t hesitate to do whatever’s necessary. If she has to fake a romance for the cameras to get sponsors, she’ll do it; if she needs to ally with someone she doesn’t want to, she’ll do it, because it increases her chance of succeeding. (Think of a lot of her thought processes in Catching Fire, where she doesn’t want to ally with Finnick but considers that Haymitch worked out something, or where she thinks about how everyone will eventually turn on one another and have to kill each other; she’s all about the Slytherin reasoning here. There’s no noble Gryffindor chivalry.) She’s always calculating and trying to figure out how she’s going to survive (and how the people she loves are going to survive), and screw everyone else. The quotes about how Katniss will choose whoever she can’t survive without, how Katniss only cares about Prim, the only person in the world she’s sure she loves, the way she’s constantly trying to work out alliances within the Games - it’s all very, very Slytherin.
And that’s why I really do think Katniss is a Slytherin at heart. She displays an immense amount of bravery, and she does tend to be a more action-oriented person: there are times where she feels more like a Gryffindor and that she lacks Slytherin subtlety - points where the reader is aware of something and she isn’t, or where she’s too blunt and not quite able to deceive (think the Capitol’s tribute interviews; she’s not good at playing Capitol word games, or pleasing people she hates.)
(As for goals, Katniss’s goals are actually strictly Hufflepuff - she wants a life of comfort and safety for her family, not fame, not power. But Katniss is definitely not a Hufflepuff: she’s not about teamwork, or love and tolerance for all, or humility; she’s loyal to her family, but she’s not defined by Hufflepuff traits - she’s not particularly humble.) Ravenclaw is right out - Katniss is intensely focused on what is; she’s not an inventor, not particularly witty, not fond of learning - not unless she’s learning a subject that will aid her in her immediate survival or further her own ends. And that, again, is a very Slytherin trait: the interest in knowledge only insomuch as it is useful.
If you’re looking at power vs. glory as Slyth vs. Gryff goals, her goals there are wholly Slytherin: she wants the power to control her own life, the power to keep her family safe.) I think you could say that Katniss is a Slytherin who’s forced into a very Gryffindor role throughout the books: she’s made to become that symbol of rebellion, the famous face, the Girl On Fire whose element is emblematic of Gryffindor - but it’s not something she wants, and she uses her position in the Rebellion to maneuver for the power she wants and needs: the power to guarantee immunity for other Victors, to kill Snow, to get a little bit more control over her own life by having some hunting time with Gale.
Katniss does have daring and nerve; she is brave and quick-tempered; she even shows a touch of chivalry in protecting Rue. But in the end, she’s more a Slytherin, I think. She’s willing to do whatever it takes to save herself and her family; she has that “certain disregard for the rules” that marks both Slyths and Gryffs, but her aims and her personality are more fundamentally Slytherin. She’s not out to save the rest of Panem - only herself and the people she cares most about. And she’ll do that through any means necessary, noble or not.
So, yes, I define Katniss as a Slytherin; if I was ranking her, I’d say Slytherin > Gryffindor > Hufflepuff > Ravenclaw. That Gryffindor streak is there - but ultimately I’d say she’s more a Slytherin with Gryff traits than a Gryff with Slyth traits. (Not all heroes automatically go to Gryffindor!)
(Peeta, I believe, is a Hufflepuff. Gale is similar to Katniss, with both Gryff and Slyth traits: they have the same inner fire and temper but also the same focus on power rather than glory. The ‘Claws of the series mostly stay in the background: Cinna, Beetee, Foxface, possibly Annie, though she isn’t very developed.)