Why was Teddy using Leonard in Memento?

When Leonard's wife was raped, nobody believed Leonard that there were two people involved. One that Leonard beat down, and a second one: "John G".

Leonard: There had to be a second man. It was from behind, I remember. It's about the last thing I do remember. But the police didn't believe me.

Teddy, the cop that was working the case of his wife that died later through insulin overdose, was the only one who believed his story.

His help started out as an act of pity and probably he also wanted to deliver justice, where the law failed.

Teddy: I'm not. I was the cop assigned to your wife's death. I believed you, I thought you deserved the chance for revenge. I helped you find the other guy who was in your bathroom that night. The guy who cracked your skull and fucked your wife. We found him and you killed him.

Unfortunately, Leonard could not remember the revenge.

I believe that Teddy actually developed a real sympathy and friendship towards Leonard and he realized that Leonard would be lost without a cause, like he was before his wife died. (Remember that parts of the story he tells about Sammy Jenkins are actually what happened to himself. His lethargy was what ultimately got his wife to kill herself.)

So he decided to help him keep his search going, mainly because he wanted to see him happy again.

Teddy: You didn't remember, so I helped you start looking again, looking for the guy you already killed.
[..] I've never seen you so happy - I was convinced you'd remember.
[..] Fuck you; I gave you a reason to live and you were more than happy to help.

He usually led Leonard to bad people, criminals. That's how he kept his conscience clean and kept carrying out his acts of self justice. I don't believe that Teddy was a corrupt/dirty cop, but that he was disillusioned.

With Leonard he could beat two birds with one stone: Help a friend to keep his life together and carry out his very own "Avenger" mission against criminals.

But somewhere along the road he got selfish and started to think: "Why not make some profit out of it?"

That's why he told Jimmy he wanted to sell drugs, so that Leonard would kill him and he would get the money.

Teddy: Jimmy's your guy, Leonard. I just figured we'd make some money on the side.

So in conclusion I think that Teddy's motives started out good and still are to some point, he actually wants to help Leonard, but he got corrupted and more and more started using him and his condition as a tool for his own benefit.

Teddy's only interest is profit.

Outside of some self-serving dialogue from the untrustworthy Teddy himself, nothing in the movie suggests Teddy was ever acting out of kindness. Instead, the movie shows Teddy trick Leonard into killing Jimmy because Teddy had stolen from Jimmy -- which is definitely not a way to "help" Leonard.

Teddy is not a nice person -- he's a scheming criminal. And after gaining an understanding of Leonard's memory problem (reportedly as the cop assigned to the case, although that could be a lie) Teddy realizes he can get Leonard to do some of the dirty work required to rip off bad guys:

"You're not a killer. That's why you're so good at it."

"Cheer up. There's plenty of John Gs for us to find."

I believe that the truth is a mix of the other two answers.

Teddy was in fact the cop assigned to Leonard's case. He did help him kill the original John G. However, after that, after Leonard didn't remember and kept searching for a man he planned to kill, Teddy saw it as an opportunity to knock out some criminals. From his perspective, Leonard was going to kill a lot more people anyways, so he might as well make them bad guys.

It's possible that there was some concern for Leonard, or that he started to think of him as a friend, but if so that still wasn't his main reason for doing what he was doing. He was a police officer who wanted a way to wipe out all the criminals he was after without himself committing murder and risking going to jail. And Leonard was the perfect way to do that.