Basically, there are two main criticisms that I've seen:
1. Chris Kyle was actually an awful person and this portrays him falsely. He called the Iraqis "savages" and said he "enjoyed what he was doing" and "liked killing bad guys."
Here's an article attacking Chris Kyle as a psychopathic killer:
http://Jimmer.it/vD0sIl
Here's a video where he defends and explains what he meant:
http://Jimmer.it/lRlLsm
2. Similar but slightly different. People are saying this is war propaganda. They say it glorifies war and makes the Arabs look evil. They say it's a flashback to old war propaganda and people are just eating it up.
Article saying that:
http://Jimmer.it/sBemX3
Another article saying that:
http://Jimmer.it/W0i6ns
So essentially, the argument is that the movie promotes war, glorifies killing and heroizes a villain who enjoyed killing. People are also saying it's trying to justify a bad war.
My opinion? I don't find Chris Kyle racist. I haven't read his book... but his interviews make more sense to me than his critics. I think he really is an American hero, and he saved a TON of American lives.
The quote where he says something like: "I don't regret killing. I will stand before my maker and account for every shot I took, my only regrets are the people I could not save," is a real quote. I believe him.
Also, I watched the movie with two Iraq veterans who both have PTSD. They said this movie is the most accurate depiction they've seen of what it's like to come back from war. They kept saying to me: "You see why we wanted to go back there? You see that we felt like we were protecting our brothers? You see the evil we saw day in and day out? You see why we wanted to stand up for the innocent over there?" Both said "Lone Survivor" VERY accurately portrayed what war is like, and that "American Sniper" portrayed what it's like to come home.
This movie made me understand them better. My grandpa, my friends and the thousands of other active military members around the world are heroes in my book. Michael Moore, Bill Maher and Seth Rogan can keep enjoying the free speech these men lay down their lives to protect.
This article also claims to have everything about the controversy, but I haven't read this one:
http://www.eonline.com/news/617496/everything-to-know-about-the-american-sniper-controversy