You'd have to have the marriage occur in one of the openings between one of Ivan's eight marriages (talk about scary). Although Russia and England first established a diplomatic relationship in the reign of Edward, Elizabeth was more concerned with commerce with Russia, whilst Ivan desired more. His interest in Elizabeth apparently dates towards the end of his reign, when he desired asylum in England, for some reason. 1580s is way too late for any kind of marriage -- Elizabeth is by then past childbearing. It'd have to happen in the early part of her reign, the later 1550s or early 1560s.
Honestly, I can't see the two marrying. Elizabeth marrying the Danish Crown Prince of the Swedish King seems much more likely. The biggest issue is religion; Elizabeth certainly isn't going to convert to Orthodoxy and I don't see Ivan coming around to Protestantism anytime soon. There is also the matter of distance; it'd be difficult for Elizabeth to effectively govern from Moscow and likewise for Ivan to govern from London... I also can't see the two really getting along. Two very dominant personalities, sounds like they'd definately clash.