League is not actually a Team Game
On the surface, League of Legends appears to be a team game. You work with 2-4 other people to succeed at the objectives, and if you do so, you win. However, this team-play itself is undermined by several core mechanics of the game. It is, in fact, a single player game with competition between each player as well as between teams.
The simplest and most obvious source of this dichotomy is 'kill stealing' and the mechanics that feed into it. Kill stealing wouldn't exist in a true team-game. In a team-environment, you can't steal a kill, your team gets credit for a partial victory.
In League, this is not the case. Specific players get kill credit, and this is based SOLELY on the kill-shot. It is further hammered in by the 'assist' tracker, which is even included after your deaths in an implicit statement of 'this matters less than anything else in the game.'
Two things could change this problem, and pull the player-vs-player mechanics back in line with team-play.
Option 1: Instead of last-hit credit, base the credit for the kill on percentage of damage dealt to the target before it died. This is a measure of 'who did the most work' and thus more fair. Even if Nidalee snipes your kill you've been battling with for 2 minutes, you get the kill because you did 80% of the damage. Nidalee would get an 'assist' for finishing it off with a tap.
Option 2: Remove assists entirely. There is only kill and death, and anyone who taps the target gets equal credit. This still measures 'who participated' in the kill, but doesn't rank them or cause a competition. Instead of 'I'll save my execute for the killshot' it becomes 'I'll need to keep my execute handy so we drop this guy!' -- people might be annoyed that someone walks in, auto-attacks once, and still gets credit, but the point is that that auto-attack can't take away YOUR credit after all your work. It incentivizes people to come and help rather than make you despise your teammates going for the 'late gank' so they get credit after you've been forced to break off. Instead, you both get the kill added to your score.
Either option could also be applied to minions to avoid creep-farm stealing between lane partners. It would incentivize both to wipe out their creeps as quickly as possible WITH THE RISK of not having their resources available for the opposing team. Risk/reward generally implies stronger gameplay, I'm told.
The increase in gold award could be mitigated by either dropping the total gold given per kill, or raising the prices of items in proportion.
Another issue with team play is certain lane roles. Junglers in particular get stuck in awkward situations with their laning teammates. Leashing adds early game issues to the laners who help, and without the leash, many jungler champions are crippled too much to do enough good in the game, even if they are cooperative and helpful.
There are two options here as well.
Option 1: share the benefits (other than buffs) of killing monsters with the team. This way the jungler doesn't lose anything to overzealous leashing, and leashing actually gives a gold boost to the laner losing time.
Option 2: make the first jungle camp attacked solo-able by any champion, then switch them back to standard after one camp has been cleared. This could be made into a competitive element, as it could be the first camp FOR BOTH TEAMS, allowing you to push a nerf onto the other team if you're fast enough, forcing them to leash to keep their jungler in the game.
These are my suggestions for improving the team-play so that League is more friendly and less toxic by default.