Of all the characters on The X-Files, Mulder is the one with whom Krycek has had the closest and most intense relationship. You could call them best enemies. Mulder's antagonists are mostly invisible and untouchable - people and organizations who are shielded by secrecy or status. Krycek is the human face of the enemy, a man similar in age and position to Mulder, whom Mulder can deal with, face to face.
Krycek was originally sent to spy on Mulder, while posing as his loyal partner. Mulder resented and distrusted him at first, but was soon won over. Krycek seemed like the perfect partner: intelligent, competent, helpful, and enthusiastic. And unlike Scully, he believed in Mulder's work. In "Sleepless," he accepts Mulder's theory about Cole's ability to create illusions. He eventually kills Cole in the belief that he is protecting Mulder. He brings Mulder coffee and takes him home to make sure he rests. Small wonder that Mulder grows to trust Krycek. By "Ascension," Mulder is taking Krycek for granted the same way he takes Scully for granted. Against Skinner's orders, Mulder goes to Skyland Mountain in search of Duane Barry. And he blithely assumes Krycek will come with him. ("Get your car, meet me downstairs in five minutes.") This seems more than a little self-centered; Mulder may be willing to commit insubordination, but demanding it of Krycek is presumptuous at best. Krycek is a new agent who can't afford to start off on the wrong foot this way; it's inconsiderate of Mulder to expect it of him. But Mulder does expect it of him, just as he expects it of Scully. And Krycek eventually betrays him, stopping the tram and keeping him from saving Scully.
It's not until Mulder borrows Krycek's car and finds one of CM's cigarette butts in the ashtray that he realizes Krycek is not who he seems to be. By then it is too late. Krycek has vanished...and Mulder is furious.
The next time Mulder sees Krycek, he beats the living daylights out of him and almost shoots him. Mulder is convinced that Krycek murdered his father. While we, the viewers, know that Krycek was at least at the scene when Mulder's father died, Mulder has no way of knowing that. He's under the influence of a psychedelic drug, however, and Scully has to shoot Mulder to keep him from killing Krycek.
But Mulder's violent behavior cannot be entirely attributed to the drug, because he beats up Krycek again the next two times he sees him ("Piper Maru" and "Tunguska"). Krycek appears to bring out Mulder's baser impulses. Mulder's actions in "Tunguska" were particularly unpleasant, crossing the line to police brutality. Despite this, Mulder trusts Krycek (over Scully's objections) and brings him to Russia with him.
In Russia, Mulder, who does not speak Russian, is dependent on Krycek, who does. And they seem to work well together for awhile. (One Usenet wit dubbed them "the dysfunctional Hardy boys.") Mulder lets Krycek out of the handcuffs and eases back on the physical abuse. Then they are captured and imprisoned in a Russian gulag; Krycek gets himself out, but leaves Mulder to be experimented on. Mulder vows to kill him, but only beats up on him a little more during his escape. "Terma" contains a couple of shockers (for the viewers - Mulder doesn't know yet): Krycek gets his left arm amputated, and we find out that he is apparently a Russian agent.
The next time Mulder and Krycek meet is in "The Red and the Black." Under orders from WMM, Krycek breaks into Mulder's apartment. He ambushes Mulder, takes his pistol, and tells Mulder (at gunpoint) he must rescue a captured alien rebel. But Mulder doesn't trust Krycek and doesn't believe in aliens any more. Krycek tries to convince Mulder of his sincerity - by kissing him. A long, audible smack on the cheek! And apparently it works, because Mulder eventually does try to rescue the alien.
This kiss between Mulder and Krycek was surprising; it's not a Russian custom, any more than it's an American one. The scene can be read in many different ways - intentionally so, no doubt. Perhaps Krycek was just trying to shock Mulder into action, or playing mind games with him. Perhaps it was the only way Krycek could think of to express his sympathy for all Mulder's gone through, or an apology for hurting Mulder in the past. Or perhaps it was an acknowledgment of sexual attraction. In any case, it gave certainly Mulder something to think about. Krycek gives Mulder his gun back and walks away, saying "Good luck, my friend" in Russian as he leaves. Though Mulder's finger tightens on the trigger, he doesn't raise the gun against Krycek, let alone try to shoot or arrest him. Instead, he sits there on the floor, staring after Krycek. When Scully finds him later, he's brooding in the dark, holding the piece of paper Krycek gave him, muttering distractedly about fate and destiny and inextricable relationships.
So we're left with another strange twist in this strange, twisted relationship. First partners, then mortal enemies, now - allies by necessity? Comrades in arms? Friends? Lovers? Alien invasions make strange bedfellows, I guess....
Mulder and Krycek do not meet again until "Biogenesis." Mulder, under the influence of an alien artifact, collapses in the stairwell of American University. Krycek is there looking for Barnes (as was Mulder). He sees Mulder lying on the landing and walks past him. Mulder looks up, apparently recognizing Krycek but too weak to do anything about it.
In "Requiem," Mulder is forced to work with Krycek. He'd clearly rather beat him up, but Skinner persuades him not to. Krycek warns him that the ship is repairing itself, and they can't afford to wait too long. To no avail - Mulder ends up abducted by the alien ship.
In "Deadalive," Krycek saves Mulder's life - perhaps intentionally, perhaps not.
Krycek may have given Mulder the vaccine. (He had the chance; he was alone with Mulder for awhile, before Skinner came in.) He had to know there was no way Skinner was going to agree to kill Scully's baby. He was playing with them, like a cat with a mouse. But did he come there, just to play with them? I doubt it. That part, including the stuff about Scully's baby, was incidental. A little fun on the side. His real purpose was to save Mulder (or at least keep him from becoming a dangerous alien).
Even if he didn't give Mulder the vaccine, it was his actions that saved Mulder's life. He drove Skinner to disconnecting the life support. And he told Skinner it was a virus - otherwise, Scully wouldn't have known to administer anti-virals.