What was Kyle really after in Money Monster?

The other answers have outlined quite well how Kyle was acting primarily out of anger and might not have thought everything through entirely or might not be entirely sure what exactly he wants. It's also clear, as pointed out in Johnny Bones's answer that he didn't even want his own losses back. He's offered it multiple times and says he doesn't care about it. Ultimately what he wants is an explanation to what happened to his money.

But I'd go a little further with this and it directly relates back to what the film itself actually wants. A primary reason for why Kyle stormed the studio in the first place, beyond gaining publicity for his concerns is that he didn't just want answers, he also wants someone to blame for it, so he doesn't have to face his own responsibility. At first Kyle picks Lee as that person, since he considers his show and his persona a primary culprit in how the financial system supposedly steals from common men like him (disregarding the fact that he willingly took his advice).

And the film very much wants this someone to blame the mess onto, too. And it, together with Kyle and Lee, finds this culprit in Walt Camby, since he is the answer to where Kyle's money went. And in this point of what Kyle wants, in my opinion the film's criticism of the financial system falls rather flat, since it is satisfied with the easy answer that Kyle is looking for. They want a scapegoat, a money monster, an evil CEO to finally admit that it was "wrong" what he did, which Walt ultimately does. Nothing else did Kyle, and ultimately we the audience, want to hear and nothing else did the movie want to show. And once Walt the money monster has been exposed, Lee and Patty can literally discuss how the show will continue next week.

And in his analysis of the film (unfortunately not available in English) Wolfgang Schmitt goes even further with this. He posits that in its search for a single scapegoat, what the film actually wants is to save the financial system. The most interesting part of the film was where neither the company leads nor the quantitative programmers knew what happened nor had any responsibility for it, it was just a "glitch" in the algorithms. There was no answer to Kyle's questions. But at the end of the day noone else but Walter Camby was responsible. He was greedy and blamed the perfectly working and pure computer algorithms for it. A straight case of "human error". It's as Walter said at the end

You only came after me 'cause I lost your money. Nobody was asking questions when everybody was makin' a profit. You just gobbled up every dollar of stock you could afford.

And if Walt Camby hadn't done shady things (or even better, if his plan had worked), investing all his money into a company because some dude on TV told him to would have been a "genuis" thing for Kyle to do.

Now it would be a bit too much to insinuate that Kyle wanted that, too, at least not on the larger scale. But he did want an "easy" way out, a greedy person to blame everything onto, not realizing that the answer isn't as easy and this way also denying his very own part of the responsibility for his losses.

Holding a public persona hostage is going to get you a lot more coverage than holding a random person hostage. By walking on the set and doing it live, he was guaranteed to get a much quicker response to his concerns.

His main focus was to get answers about how Lee could be so wrong. However, over the course of the hostage crisis, Kyle realized that things didn't add up. I wouldn't say he didn't think it through, as stated in another answer. I think he was aware that something could go wrong but didn't care. He really didn't have much to live for. His girlfriend was overbearing, he was broke, he acted more out of desperation than anything else.

I don't think Kyle was looking to recoup his losses at that point. IIRC, he was offered more than his losses. However, once things started looking shady, I think Kyle began acting more like a "guardian angel" to other investors. He was going to get to the bottom of the stock crash no matter what the cost, as if being that guardian angel justified/defined his life.

It was an act of anger. He had lost all his money and he held two people responsible for this. Lee and the CEO of Ibis. However, he did not think it through as is implied from the fact that he used clay blocks instead of actual explosives. His motives were not to kill anyone but just to get his money back. An action acted up without proper thought and preparation.

The anger that he holds is also out of that fact that the two people that he thinks responsible are still rich while he is still working on his low paid job and has lost all his money. So the whole "hostage" situation arises from him acting without proper planning, hoping to recover his money somewhere in the process.

You can see most of his actions are either unplanned (due to the situation) or impulsive.