Guys, upgrading parts wont do much since his motherboard will be in the same generation of cpu sockets, ddr2 ram is outdated as ddr4 has come out into the public, and I doubt that board will have much support as far as PCIe slots for a decent graphics card.
It would be much better to start from scratch on a new build really.
League doesn't require much, so you actually don't need a card, an Apu from the fm2 series you could keep it at a decent gaming computer price.
Your best option, and I'm assuming that since you have an e machines you will be on a budget?, is that building your own computer would be best. Since I will come up with a build for league for you, all you would really need to know is how to shove wires where they need to go and such. There are many Youtube videos that you can learn how, and even follow along when doing it yourself need be, to help you build your own. Since all the parts are coming from newegg, Ill link their video tutorials.
https://www.youtube.com/watch?v=d_56kyib-Ls
https://www.youtube.com/watch?v=RxaVBsXEiok
A build that would handle league at 60fps:
cpu/gpu(this is an apu, it basically combines the two, you lose out a bit on performance, you can save on power and money in many situations)
55$
http://www.newegg.com/Product/Product.aspx?Item=N82E16819113335
motherboard:
75$
http://www.newegg.com/Product/Product.aspx?Item=N82E16813132115
RAM(random access memory)this is not storage as was clarified by earlier posts
30$
http://www.newegg.com/Product/Product.aspx?Item=N82E16820231277
Power supply 600Watt will allow for a bit of leeway if you ever decide to upgrade, it also allows for degrading in this build.
70$
http://www.newegg.com/Product/Product.aspx?Item=N82E16817139048
Case, just a small micro atx case, but the design allows for room if you ever want to get a discrete graphics card.
35$
http://www.newegg.com/Product/Product.aspx?Item=N82E16811553011&ignorebbr=1
Storage( going with less than 500GB is actually the same price, so may as well take it)
50$
http://www.newegg.com/Product/Product.aspx?Item=N82E16822136769
The most expensive part is an operating system, if you go for windows that is. If you were to build this soon, you could go for win7, and take your time to see if you want to upgrade straight to win10 (microsoft is currently offering free updates from 7/8.2 to 10 to all existing customers.) Of course if you want you can just straight up buy a version of Win10, and not have to deal with the mess of upgrading.
Win7 will still receive updates though, if you prefer to stick around with an operating system you may know your way around better.
of course this build does not include a Optical drive, and I can't seem to find out if the old one will work or not. Anyways, this is a usb port version of win 10.
Now this was a 315$ build plus 100$ for windows if you don't have a spare product key laying around. This is about as cheap as you can make a build.
Once you get this together, or if you have the extra money I would first look to doubling that RAM(memory) up, as 8GB will( and already is) become the standard.
Next I would look to getting a bit more cooling on the apu, as we aren't able to really fit an aftermarker cooler in because of the smaller case, getting some fans can be a good step.
Your case will come with 1, and can fit 3, so you can buy 1 or both when the time comes.
http://www.newegg.com/Product/Product.aspx?Item=9SIA24G28M8124
