open the LOL folder in your c: or what ever drive you have it installed to
Riot Games > League of Legends > air > logs
open your most recent log file with notepad
do a ctrl+f for "ELO ="
1200 is the base ELO everyone starts with I guess 50/50 or whatever.
So as you get worst or better your ELO will fluctuate below and above that number.
Note that this only works for the first game recorded in the log and if the first game is in normal mode (because you can already see your ELO in ranked games).
So if you want to get a recent one, start up LoL, play a normal game, and close the client. Then open up the most recent log and follow the steps above.
The structure of the Log-file is (about) as follows: your elo
elo of the opponent team's players
elos of your team's players
so if u type in "elo =", the first elo that shows up is yours