A few thoughts on your question.
First, the specific concept of "moral responsibility" is foreign to Aristotle's ethics. But it's not completely foreign; instead, he's going to look at people and their actions in terms of praise and blame. He addresses voluntary action in BK III of the Nicomachean Ethics (hereafter NE). On his picture, ethical action must be voluntary choices built chosen freely and built on deliberation.
Second, Aristotle specifically mentions poor upbringings and bad circumstances in several places in the NE (for instance I.10). But these do not affect whether or not a given action is ethical. Similarly, as you note, Aristotle suggests that only someone who is raised well can act ethically on his account.
But it does not follow that someone who is not raised well or incapable of discerning the good is then excused from blame when they fail. To understand this, we need to recognize that Aristotle's ethical is not the same as Kant's moral. I'll skip over many of the details of Kant's views but a key feature for the purposes of this point is that "ought implies can." In other words, for Kant, everyone has the power to act morally but not everyone chooses to act morally.
Aristotle's picture is different. Sometimes we distinguish this by being careful about using "ethical" instead of "moral" (regrettably, these two terms are used in many different and confusing ways). But in Aristotle's case, the idea is this: the good action is the one that reflects correctly human nature and is the wisest action that mirrors this (building on the function argument in NE BK I.7). In other words, the standard is not whether you personally can do the action but rather what a human being trained and reasoning appropriately would do in those circumstances. If it is the case that you cannot do, then you will act badly (which for Aristotle means you will be blameworthy). In other words, his account lacks "ought implies can" and with this lacks any sense in which the inadequacy of one's circumstances or prior training legitimating or justifying a wrong action.
Aristotle makes this point explicit in NE BK I.8 (and again at I.13) where he maintains that to truly flourish, you will need (a) external goods and (b) proper upbringing. He repeats the point in BK VIII where he identifies friendship as necessary to our lives. Also, he thinks a certain political structure is necessary for human flourishing (BK IX).
In an answer to your specific worry,
Say, I have a poor education and I am not taught the right way, and so by no fault of mine have a very different notion of righteousness from what most people have. When I become a fully grown adult, I strive for bad ends, performing bad actions as means to those ends. Should I really be blamed for my actions?
For Aristotle, none of this abrogates (eliminates) blame. The fact that you're upbringing did not enable you to be ethical doesn't for Aristotle exempt you from the requirement to be ethical.
This is one of the harder points to accept in Aristotle and some contemporary virtue ethics seeks to soften this point, but Aristotle's own language is not ambiguous on this point.
Put another way, if the task of excellence were to be running a mile in 4:00, then there's only blame if the best you can do is 6:00 and no time to listen to your excuses about why you cannot do it.
This is harsh, but Aristotle's ethics doesn't have the same virtues as later Christian and post-Christian ethics. Mercy is not a key value of the ancient Greeks nor is allotment for disabilities or poor upbringing.
(In the above count, I've skipped over two things that I want to mention now.
First, the example with running a mile in 4:00 is not ideal -- but it's not ideal for reasons that are immaterial to answering the question here. One problem is that Aristotle's ethics are adaptive in that the mean is excellence with respect to your own nature, time, and circumstances. But, this does not mean that Aristotle adjusts this to the point where someone disabled is going to reach his bar of excellence.
Instead, it's something that raises a far more complex problem. Specifically, to know what is most excellent is something that is limited to the man of practical wisdom (phronemos), and those who are not phronemos are not going to be able to find the exact mean of excellence for action).
The second issue is that there's some question as to whether actions or people are the one's who receive praise/blame in Aristotle or how the two relate if both are involved (as they seem to be in his account).