If you go to na.op.gg and look yourself up before a game starts, you can click in-game info. From there, it's basically like lolnexus if you've used such things. From there, you can click an option that says 'record' or something like that. It's a grey button with a camera on it.
Sometimes, the server won't let you record because it's busy or something like that. But usually, it records for me. If you want to watch your replay, just refresh your match history on op.gg and the match you recorded will have a small 'replay' button of sorts on the top right of it. Just click it and it'll attempt to save it onto your computer. It's a safe download, but if you use chrome, it'll say that the download is harmful.
To override this, go to your downloads (ctrl + j), and select the op.gg download and click the override option. It'll download and you can select it from your computer and watch your replay.
Sorry for the long explanation, but hopefully it'll be useful for you.