I am not sure that you can view someone's profile without being their friend. But there is some information you can gather on them.

If you go to lolbase.net you can search their name and it will show thier level, how many wins they have and also how many leaves. You can look at their past games and see how they built certain champs they have played.

Note: Lolbase.net gets its info from players who upload their logs after games. So if someone has uploaded logs or played with/against someone who has then their name will show up. But if they havn't uploaded their logs or played with or against someone who has, you won't find any information on them there.

Hope this helps

In your friends bar, the middle icon on the bottom looks like a magnifying glass. Click it and you will have a little search prompt that will let you see a summoner's profile. Remember that you are only (supposed to be) allowed to see their wins.