Each week on tuesday Europe servers there are 10 free to play champions. Those can be looked up here. If you want to get the champs for free without paying for them you can use your influence points to buy them. But I suggest you try and play the champion first before you buy. Else you will be stuck with some champions you don't like playing, and will probably never play.
At times aswell if you want to throw some real cash to champions, Riot offers cheaper rp champions, only three or so.
As Raven Dreamer said there are only 2 champions that you get for free : Tristana and Alistar.
But my friend got myrmidon Pantheon for free on a smurf account he didn't play anymore, they gave him a key (to get him playing again), saying he will get the free skin and he just used his code on his main account. (Don't know if they still do this)
Not to mention if you play long enough till the end of season 2, and you win either gold or platinum you will get a free skin and if you just play 10 matches (Summoner Level 30 needed), you will probably get one too... atleast I hope they do that again