Your ranked Elo should be visible once you've played five ranked games (you'll still be in the ranking process until you finish ten, but you can see how you're doing once you finish five). It's visible in a lot of places in game, if you're in a chat room, you can mouse over your icon to see it. If you click on your summoner profile (the place where you set your rune pages and masteries up) you can see your Elo under the ranked stats section, I think. It should give you a few different Elos for the different modes you've ranked in (solo queue, 5v5, etc.).
If you're not in game, you can go to lolking and type in your summoner name and it'll look it up. If this is you:
then your Elo is 1275 right now.