Quick way to get 7-8k BE on a new account

At level 5 receive you a capsule with 3260 BE and a random token (high chance of Caitlyn) this effectively allows you to purchase any champion at level 5.

And I also recently found out that you can get 3000 Blue Essence from completing the tutorials. It's 3 quick games that take less than 20 minutes to complete.

If you do this you'll be able to buy whatever expensive champion you want at lvl 5!