I want to point out first that what I'm sharing here is information that is common to most tribes, but this is very basic information. If you are writing about a character from a particular tribe, it's a good idea to either talk to someone who is involved with the spiritual traditions of that tribe, or find a good written source by someone who is a member of that tribe to add the more specific details. However, having the basics gives you something to work with in the event that for some reason you can't find a good source.
One of the most basic tenets of Native American spirituality is that everything is alive and has spirit and as such is deserving of respect. This includes such things as non-Indians consider to be inanimate objects or not alive such as stones for example, or rivers, or oceans. In our traditions these are beings and have consciousness and can interact with us if they so choose. It is thought these beings have a closer connection to the Creator and the Spirit World in general and can even relay messages to us. Also, they have a different POV as it were and can give us information based on this that is helpful.
This is one reason you will hear NA's call the stones that we use in the sweat lodge "grandfather" or "old man." These stones chose us, we don't choose them. This is a hard concept to explain but when we're looking for stones for the sweat lodge it's not just a matter of going out and picking up the first rock we see, it has to be a particular type of rock, for example, you would never use a river rock because they would explode in the heat and could injure people. Lava based stones are what you want for the sweat lodge because they can handle the heat. But in addition to that, the stone has to want to participate. I've been at sweats where stones have been taken from the fire on a pitchfork, and the rock will just roll or fall off the pitchfork. We know that if the person picks it up again with the pitchfork and it rolls or falls off a second time, it does not want to go into the lodge yet, and the stone is put back in the fire until later when another attempt may be made. I'm using this example because it illustrates our understanding that these stones are far older than we are and in our understanding are sentient.
While some animals are considered more sacred than other by different tribes, although all animals in and of themselves are sacred and have power, it is a nearly universal belief that eagles are messengers from the creator and as such are very sacred. This is one of the reasons eagle feathers are used so much in our art, and regalia, and why you may see our dancing and note some of the very bird like gestures. This is why if we notice an eagle circling an event such as an outdoor powwow or a sweat that this is considered a good omen so to speak.
Now I've had a lot of people ask me to explain whether or not Indians believe in "God." The best way to answer this is that our god concept is totally different from Non-Indian's god concept whether that person be Christian, Jewish, Muslim or whatever. My understanding is that our spiritual beliefs with regard to any sort of divinity are closer to other pagan systems, even the newer ones such as Wicca than any male-based system. In our traditions, there is a Creator but this being is neither male nor female, nor is this being human in any way (no old men with flowing white beards here, or even beautiful women for that matter), nor does this being have human emotions like anger, or jealousy such as you read about in some religions. I don't say this to knock other religions but simply to illustrate the differences.
So, we have a Creator being that I guess is best explained as being at the top of the heirarchy, although honestly I don't think there really is a heirarchy but for the sake of trying to explain this in English which is tough, I hope that helps your understanding. Then there are other spirits and beings such as the Spirits of the Directions, Mother Earth and Father Sky. Finally, there are very highly evolved spiritual individuals who walk among us that are sent by the Creator to teach us, one example of this is White Buffalo Calf Pipe Woman who brought the pipe and several of the ceremonies that are the core of traditional Lakota spiritual tradition, among the Nez Perce there was a man named Rainbow that was equivalent. Both White Buffalo Calf Pipe Woman and Rainbow would be to us as Christ would be to someone who was Christian. And of course there are others depending on the tribe. When you include all of these with the fact that animals and stones, trees, rivers, oceans, plants etc. have spirit and are also part of our spiritual universe and in connection with the Spirit World, I think it becomes clear why I liken us as being closer to pagan traditions.
This should give you a pretty good overview of Native American spirituality, but as I've said, there is much more detail depending on your tribe, and it's specific spiritual traditions. If you're going to write authentic NA characters, then you will want to include some of this basic information, unless of course you're writing an NA character that doesn't have a connection to his roots, but even then you may want to consider using this information to lead him or her back to it. I've seen this done pretty well in a few fiction books. The unfortunate truth is there are NA's with no connection to their roots, so it's not completely unheard of, so writing such a character isn't necessarily bad.
When writing NA characters becomes bad is when someone begins making things up rather than doing any research. I can give a great example, back in the 80's there was an author, and I can't for the life of me remember her name now but she wrote those historical romances where the white woman is captured by an Indian chief, yeah I'm sure you know the ones I'm talking about. (rolls my eyes) Anyway, she had the woman captured and then more than once if I'm not mistaken, her "husband" ties her up and beats her in front of the entire tribe. Now first of all, yes, there were bad Indians and good Indians within the tribes, just like you have bad and good people now. That's just a fact of life, but the writer made it sound like this was normal behavior because she wasn't submissive enough to suit her husband. I was fifteen or sixteen when I read that book, and it about made me gag. Generally speaking even in tribes that are more patriarchal such as my own Lakota, women are revered, they are considered sacred themselves because they give their blood back to the Mother every month, and they can give birth which is also seen as sacred. These men did not beat their wives, even their white ones in most cases. Are there exceptions, sure, you had bad guys, but even were it to happen, this behavior would not be tolerated by other NA's at all. She would have been taken away from him in most cases, and he possibly could have been banished. Women could and did and were allowed to have their own thoughts and speak their own minds. They played a major roll in the decision making for most tribes, and in some cases as with the Iroquois for example who are matriarchal in the extreme, they had the final say. This particular author took what she knew of stereotypes, and her own thoughts about a woman's place in the 1800's and projected them onto her characters. Very very bad move if you're trying to write authentic NA characters.
Now if you're looking for a fantastic example of an author who writes authentic NA characters, besides myself I mean LOL, meaning she uses both her imagination and her personal knowledge to write her books, read a book by Lindsay McKenna. Lindsay writes sweet romances, but when she writes NA characters, she nails it every time as far as being authentic, of course she grew up in a Cherokee family that still practiced their traditions and she's been exposed to other tribe's traditions and participated as well, if I don't miss my mark. She writes about the Lakota very authentically telling me that she's talked and spent time with our elders. While personally I prefer to both write and read more erotic romances, when I stopped reading some other sweet romance authors, I kept reading her because of the fact that she can write one heck of a NA storyline and characters.