Basically any modern laptop can play League of Legends, it is not a very demanding game. Even an Intel HD 4400 or HD 5500 will be able to run that game at 1080p with very high graphic settings and still be able to get pretty decent frame rate. Below is a video of LoL being played on the Surface Pro 2.
https
/www.youtube.com/watch?v=EPOBeiNhVnA
A quad core i7 CPU is great, but since it produces a lot more heat and consumes 3x more power than a dual core i3/i5/i7 CPU battery life will be somewhat short. LoL can easily be played on a dual core CPU as demonstrated by the above video.
As for a 13" laptop, very few of them will have a dedicated GPU; one is the 13" Alienware (~$950) and another is an Asus Zenbook (can't remember the model #, but costs at least $1,100). But both costs more than what you want to spend.
The least expensive 13" laptop is the Asus Zenbook ux305 for $700, but it has a Core M CPU which is a bit less power compared to the Core i3. Its integrated Intel HD 5300 is between the performance of the Intel HD 4000 and HD 4200; still enough to play LoL, but you will need to lower the settings. It has a 1080p IPS screen and weigh in at 2.6lbs.
Review:
https
/www.youtube.com/watch?v=Yve9TB-fXV8
The next least expensive 13" laptop is the Core i3 version of the Dell XPS 13 for $800. Due to the thin bezel around the 13" screen, the laptop is more like the size of a 11" - 12" laptop. It has an intel HD 5500 graphics core so playing LoL should not be a problem. It has a 1080p IPS screen and weigh in at 2.6lbs.
Review:
https
/www.youtube.com/watch?v=sP6oTd_OhoA
You can actually purchase the Core i5 version of the Dell XPS 13 for $799. Normal price is $899, but use the promo code
PCGRAD15 to bring down the price. The Microsoft store does not carry the Core i3 version. They also sell the Asus Zenbook ux305 use the same code to bring the price down to $599.
http
/www.microsoftstore.com/store/msusa/en_US/pdp/Dell-XPS-13-9343-Signature-Edition-Laptop/productID.312471000
http
/www.microsoftstore.com/store/msusa/en_US/pdp/ASUS-Zenbook-UX305FA-USM1-Signature-Edition-Laptop/productID.313303100
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If you want a dedicate GPU in your laptop, then consider the 15" ASUS Laptop X555LB series laptops. As far as I can tell those are the cheapest laptops with a dedicated GPU (940m). Below is a link to Newegg.com with the Core i5-5200u and i7-5500u versions.They are 5.1lbs.
http
/www.newegg.com/Product/ProductList.aspx?Submit=ENE&DEPA=0&Order=BESTMATCH&Description=asus+940m&N=-1&isNodeId=1