What does Colonel Slade mean when he asks: "Should I adopt you or shoot you?"

You need to look at the entire conversation to understand it:

[Charlie comes back to the room to see Frank putting on his uniform]
Lt. Col. Frank Slade: You're back too fast. You didn't get my cigars, did you? Get outta here, Charlie.
[loads the .45]
Charlie Simms: I thought we had a deal.
Lt. Col. Frank Slade: I welched. I'm a welcher. Didn't I tell you?
Charlie Simms: No, what you told me was, that you gave me all the bullets.
Lt. Col. Frank Slade: I lied.
Charlie Simms: Yeah, well you could've fooled me.
Lt. Col. Frank Slade: And I did.
[pause]
Lt. Col. Frank Slade: Charlie, how you ever gonna survive in this world without me?
Charlie Simms: Colonel, why don't you just give me the gun, all right?
[Frank picks up the gun and points it at Charlie]
Charlie Simms: What are you doing?
Lt. Col. Frank Slade: I'm gonna shoot you, too. Your life's finished anyway. Your friend George is gonna sing like a canary. And so are you. And once you've sung, Charlie, my boy, you're gonna take your place on that long, grey line of American manhood. And then you will be through.
Charlie Simms: I'd hate to disagree with you, Colonel.
Lt. Col. Frank Slade: You're in no position to disagree with me, boy. I got a loaded .45 here. You got pimples. I'm gonna kill you, Charlie, cause I can't bear the thought of you SELLIN' OUT!
Charlie Simms: [now firm] Put the gun down, all right, Colonel?
Lt. Col. Frank Slade: What? You givin' me an ultimatum?
Charlie Simms: No, I'm... Lt. Col.
Frank Slade: I give the ultimatums!
Charlie Simms: I'm sorry. All right I'm sorry.
Lt. Col. Frank Slade: It's all right, Charlie. You break my heart, son. All my life I've stood up to everyone and everything, because it made me feel important. You do it... because you mean it. You've got integrity, Charlie. I don't know whether to shoot you or adopt ya.
Charlie Simms: Not much of a choice, is it, sir?
Lt. Col. Frank Slade: Oh, don't get cute now.

As we know, Frank Slade is a short tempered alcoholic, whose is also blind and medically retired. He's not happy. He's on one last trip to enjoy himself, before killing himself. Charlie meets him by taking a temporary job to look after him to earn some money. Charlie is only a student, compared to the old, retired man.

This is one of these stories about understanding the good of life and how every day can bring hope and a new adventure. Charlie is so naivé in the world, so good inside compared to the bitter, cynical nature of Slade.

Charlie realises Slade is going to kill himself and rushes back to the hotel to save him. Charlie states how he really was fooled and thought Slade had gotten rid of the bullets from the gun. Slade can't believe how young and foolish he can be (which is why he asks him how he'll survive without him).

When Slade says he'll shoot him, he refers to the fact he and George are going to "sing like a canary". If you remember, this is a reference to the start of the film:

Charlie and George Willis, Jr., another student at the preparatory school, witness several students setting up a prank for the school's headmaster, Mr. Trask. Following the prank, Trask presses Charlie and George to divulge the names of the perpetrators. Trask offers Charlie a bribe, a letter of recommendation that would virtually guarantee his acceptance to Harvard. Charlie continues to remain silent but appears conflicted.

Slade is effectively telling him that once he accepts the bribe, he's going to be like everybody else. Nothing special, nothing good. Ruined.

Charlie tries to be firm with him to turn the situation around and when that only appears to aggravate the situation, he apologises - despite being the one with a gun pointed at him.

It's at this moment Slade mentions the line you quoted. What he means is that he respects Charlie a lot. Part of Charlie frustrates him beyond belief - the naivé little boy who can be fooled so easily ("whether to shoot you"), and another part of him respects this incredible kid, who'll stand up for what he believes for no reason other than because he feels he has to ("adopt you").

He's showing his conflicting emotions, that's all. Part of him is frustrated, but a larger part (which we see in the remainder of the movie) is moved by Charlie's integrity and strength of character - so much so, that Slade unexpectedly passionately defends him at the disciplinary committee and appears at the end of the film to turn his own life around based on his experiences with Charlie.