I. The Persians

This is a good question because it absolutely was an important part of Persian imperial thought, but exactly how is not well understood. I'm also assuming that you mean the Achaemenid Persian dynasty contemporary with Classical Greece based on your use of the word/spelling Aryan. In the much later Sassanid period being Eran was still an important concept, but it had about 1000 years of continuous shared political history to link all of the different Iranian groups together. The specifics of your question also have a fair amount of misconceptions, which is not a bad thing by any means but is indicative of some of the problems with how people tend to talk about "ancient Aryans." That goes for scholars and laymen alike.

I always feel obliged to start with the word Aryan on a topic like this. It's about as tainted as a word alone can be, especially in the west. That's the result of a long history of European appropriation and misuse of the term, culminating in its use by the Nazis. Academics in Iranian studies go both ways in response to this problem. Many have try to use Aryan in its genuine historical sense as an ancient ethnic name. Others prefer to translate it as "Iranian" as much as is appropriate.

Personally, I loosely fall into the latter camp even though it seems to be the minority. In my own experience, it's long misuse in Europe has led to continued similar misuse by some nationalist groups in and around Iran. I describe this as "loosely," because I'm not particularly opposed to using Aryan in the context of directly quoting a primary text. In my view, Aryan is a transliteration of an ancient word whereas "Iranian" is an appropriate translation of the same. I mostly wanted to pre-emptively address this in regard to how I'll use "Iranian" and/or Aryan in the rest of my response. So here we go:

they wouldn't stop talking about it in official documents.

Not really, or at least not in the documents that we have. It was absolutely important, but that importance is really displayed in the limited contexts where we see the Persians actively referring something as Aryan. Across all of the administrative records and royal inscriptions it's used less than 20 times, and most of those instances are as part of a personal name. The frequency that some variation of "Arya-" appears in Persian names points to some level of importance, but the exact nature of that importance makes more sense in the context of inscriptions. There's a whole system shorthand for those, see this list of links.

Aryan is only used in the Behistun Inscription (DB), Darius I's tomb (DNa), once at Susa (DSe), and the Daiva Inscription (XPh). Of those, three are the product of Darius I and all three make reference to the same events: the series of uprisings/civil wars that accompanied Darius' ascension to throne via coup. Those events are explained in DB and referenced by DNa and DSe. XPh refers to a rebellion in an unspecified part of the empire under Xerxes I, but there religious terminology used suggests somewhere in the eastern (ie Iranian) part of the Empire.

DB is the only real Achaemenid war monument, but these aren't the only references to warfare in the royal inscriptions (though admitted it is between 40-80% of them depending on interpretation). In fact, they're not even the only references to the wars early in Darius I's reign, DZc also references the Egyptian theater of the same conflict. The thing that sets these four texts referring to Aryans apart is that they are the only extant references to rebellions among Iranian peoples. Specifically, that included the Persians, Medes, Sagartians, Parthians, Margians, Saka, and Elamites (who the Achaemenids considered as Iranian). Being Iranian was stressed specifically in the context of the right to rule other Iranians. Which leads into:

Did they see them as kin or related peoples, or as foreigners/barbarians?

The Persians saw the other Iranian peoples as both related and foreign. You could compare it loosely to how the various Germanic peoples of the Holy Roman Empire viewed one another centuries later, culturally, religiously, and linguistically related with their own regional dialects and traditions.

In general, it doesn't seem like the Persians had an idea of "barbarians" in the pejorative sense. There were Iranians and non-Iranians (An-Aryans) but plenty of non-Iranians like Babylon or Egypt were highly respected while the nomadic Saka and bandit hill tribes were still considered Iranian.

One noteworthy element of XPh, DB, and probably the heavily damaged DSu is that only Iranian groups are condemned for not worshipping the supreme deity Ahura Mazda. In examples like Babylon in DB or Egypt in DZc, there's no evidence that they acknowledged Ahura Mazda at all, but they get off without critique. It seems the religious expectations were different for Iranian peoples.

What was the Persian relationship with them like?

So far, everything I've said might give a sense of Persian-supremacy. I don't want to over-emphasize that. Moreso than their Iranianness, Achaemenid ideology stressed their right to rule as Ahura Mazda's chosen king's and their dynastic legitimacy. Ahura Mazda is referenced in almost every inscription more than a sentence long and by the time of Artaxerxes III he was listing out all his paternal ancestors for 7 generations. Royal monuments routinely declare some version of "Ahura Mazda created the earth and gave it to [King]." There's no evidence to suggest that Iranians faced any more hostility as a result outside of rare religious disputes.

Modern Afghanistan specifically was split between several Achaemenid provinces representing different major Iranian groups, all likely home to smaller local ethnicities as well. The northeast was Gandara, where you'd probably see more Indian-oriented languages like Classical Sanskrit and related Pakrit languages. Bactria was in the northwest and had its own language, Bactrian.

Southern and Central Afghanistan was divided between several provinces none of which have specific linguistic associations, but were probably home to the early Zoroastrian community that actively spoke Younger Avestan, which is the language of most of the sacred Zoroastrian hymns and prayers in the Avesta. That language was fading out by the mid-Achaemenid period and its not clear if it died out or just fell outside of the historical record until people were calling it by another name. If the latter is true, then Pashto is probably the best contender for its direct linguistic descendant.