Pretty much none, Lenin's death isn't all that suspicious when you realize he's had three strokes before his death. Lenin's health had being declining for years since an assassin shot him in the neck in 1918, compounded by the stress of leading a revolution and fighting a civil war. His third stroke left him mute and bed ridden and effectively left him too sick for active politics in any case. He died less than a year after his final stroke. Under such circumstances there would have being little need to kill him.

A reading of Robert Service's biography of Stalin suggests this is really unlikely, and the opposite might even be true. Lenin's late opposition to Stalin being General Secretary wasn't about his trustworthyness, but his being 'rude'. Ironically this was because in overseeing Lenin's convalescence, Stalin had clashed with Lenin's wife about her allowing the very ill Lenin to over work. Also Lenin did not state a preference for Trotsky - his 'testament' pretty well rejects everyone for one reason or another.

Service also states Lenin asked for Stalin's help in commiting suicide, and Stalin refused. If true Stalin could have killed Lenin and chose not to. Keep in mind also that Lenin asked him to do this, and that Stalin was overseeing his care, shows Lenin and others had actually trusted Stalin enough to do this.

Now as to Stalin having a good motive - not really. With Lenin away and out of action Stalin and his allies were running the show anyway. Lenin being alive would give then time to dig in and prevent anyone else having the top job. Stalin did capitalise on Lenin's death to some extent but that was more about his statements assuming his (supposed) ideological legacy, than any practical bump in power.