Well, he wanted to wreck the Fortress of Solitude anyway and apparently has been tracking it for awhile. Personally I took it as Lex going there for other things and rather he just happened to find this. It wasn't really part of his plan yet, he just saw an opportunity he couldn't pass up. However, that raises an entire question of why he didn't do that AFTER opinions of Superman were already in the gutter to really seal the deal, but I guess by that point he got what he wanted anyway and there was no need.
What I guess really bugs me is that from Lex's perspective...He's 100% right. Superman was sent down to conquer them. Just because he didn't do that, doesn't invalidate Lex's hatred for him and genuine concern. He could turn on them at any moment and he just intervened in geopolitics, meaning he's absolutely willing to interfere in these matters. The government was entirely wrong and should've placed restrictions on Superman a long time ago. Heck, Jor-El's message even raises a good point. What if Superman DID have children? From Lex's perspective he has no reason to believe a son or daughter should inherit Superman's sense of righteousness. It's entirely possible one of Superman's offspring could and would maybe hear that message and rebel against his dad and try to the conquer the earth because "Fuck Dad" and all.
I didn't really think about it from Lex's POV before. Fuck, that message not being fake really snaps the movie in two. I guess this is what happens when you turn Superman into Goku.