What you are trying to express is a concept that is actually fairly unusual around the world in how the concepts of nationality, heritage, and identity are used. In America, Canada, Australia, and maybe other places with similar large immigrant populations, each of these concepts are options that can be combined to suit individual tastes.
In countries with longer, continuous histories and more homogeneous people, these concepts are merely different aspects of the same thing.
Japan is one of those places where culture, heritage, nationality, language, and "race" are generally understood to be the same thing. Even as Japan has become, and continues to become, more and more internationally aware, you need to appreciate that Japanese culture on the whole is still relatively early in the process of transitioning into the understanding that these things can be separate. A lot of people you encounter will be unfamiliar with the concept of what you are trying to convey.
In short, what you are trying to express isn't a matter of merely finding the right words. The concepts that drive the meaning behind the words are different between your culture and Japanese culture, and so you need to navigate that divide if you want to be understood.
Starting with your proposal of telling people you are アイルランド人{じん}, this will make Japanese people think you are from Ireland, full stop. Adding 人{じん}, in the context we are talking about, expresses "person from".
What you're trying to say, I think, is アイルランド系{けい}アメリカ人{じん}. An American of Irish descent. This gets across the point that you come from an Irish family, but I think you need to appreciate how this gives the impression that you feel quite separate from America. In Japan, as you may know, there are groups of people in Japan called "zainichi" (在日{ざいにち}). The term usually is shorthand for Koreans born and raised in Japan, but it can apply to other nationalities under similar circumstances. It is a huge political issue, so I don't want to get into it too deep, but the main point is that these people aren't merely seen as being Japanese of Korean descent, even though they may only speak Japanese and have never been to Korea. They are seen as Koreans who are in Japan, but for a variety of politically heated reasons, are not really people of Japan. Similarly, you might be seen as an Irish person who, for some reason, was in America, even born there, but not really American by either your own declaration or that of other Americans.
Going further, to express your Native American, French, or any other heritage, you could start splitting down to fractions to express how your family is delineated by ethnicity, but that would sound as overly specific and obnoxious in Japanese as it does in English. Also, I'm guessing, you are looking for a more offhand way to throw it into the conversation, not how to describe a whole family tree.
To say you are "descended from", I believe the correct verb to use is 引{ひ}く, which means "pull", so, in essence, you are "pulled from a blood line". So to say your family was of French and Native American descent, you could say 家族{かぞく}はフランスとネイティブ・アメリカンの血{ち}を引{ひ}く. "My family is of French and Native American heritage." But that makes it sound like your family is entirely French and Native American. What you want to say is that you just happen to have a little of that mixed in with your predominantly Irish bloodline. In that case, you want to use 入{はい}る, to mean the blood was "put in". 家族{かぞく}はフランスとネイティブ・アメリカンの血{ち}が入{はい}っている You can throw in a ちょっと to make it sound even more like there was just a little bit.
Bringing it all together:
（私{わたし}は）ちょっとフランスとネイティブ・アメリカンの血{ち}が入{はい}っていたアイルランド系{けい}アメリカ人{じん}
I'm an Irish American with some French and Native American heritage.
Having said that to the Japanese person you're speaking to, you still might have to contend with the fact that impression you are trying to give might be different from the impression your Japanese listener gets. That's a function of the cultural divide I've outlined above, and that can't be helped. You'll almost certainly have to have that conversation no matter what words you use to start with, but I think that at least the ones I've provided start you at the most accurate place possible given the language options.
Just as a pointer for further learning, other words you might look at are 血筋{ちすじ}, which means "lineage", but I believe is more about hierarchy, as in coming from an elite family. There's also 混血{こんけつ}, which means "mixed heritage", but literally means "mixed blood", and since it's in opposition to 純血{じゅんけつ}, "pure blood", can be used in a derisive sense.
Hope that helps.
