How do you earn more RP in League of Legends?

Solution 1:

Purchase it digitally from the in-game store, or via gift card from a variety of retail locations, such as Best Buy using actual currency.

Solution 2:

Riot Points: What, and How?

Riot Points are the "premium currency" in League of Legends, and the main method of obtaining it is by purchasing it with real world money, in the in-game store. Many game retailers will also sell Riot Points as gift cards. I would provide an image and values, but the cards and values differ, per country. You can find an official list of RP card retailers here.

The LoL store, where you can buy Riot Points

The League of Legends support page, detailing the Reward System (IP v XP v RP) states:

RP (Riot Points) can only be received by purchasing these points in the League of Legend stores. You may use RP to purchase most things in the store except for runes.

IP (Influence Points) can only be received by playing games in League of Legends. IP can be used to purchase most things in the store except for champion skins, ward skins, and summoner icons.

Other times a player may receive RP

The League of Legends Wiki has a few extra notes, concerning times where Riot has issued players Riot Points:

All players are given 400RP on reaching level 3.
Riot have been known to issue RP, from time to time, as a means of compensation. The example given is server issues, that effected players globally.
The wiki also states that, in December 2011, all players who had not been banned for the last several months were given 450RP. This was likely a "Christmas bonus", but one can only speculate as to whether Riot will offer the same bonus, in the future.

The idea behind RP

To give insight, this is a system often used as an alternative for making a particular game "pay to play". Players pay to unlock champions early, or to unlock cosmetic items. This way, the developers still make money out of their game, without directly asking each player to pay for it. Only cosmetic items require Riot Points to purchase. Champions can still be bought with Influence Points, and there is a rotation of "freely available" champions, to give players an opportunity to try different roles.

A word of warning:

There are many fake websites offering "Free Riot Points". These websites are almost always against the terms of service for playing League of Legends, and often give false promises of free points in order to steal player and account information.