These answers are less than accurate, You can totally block an IP address, You can completely handle this. First things first, Find his IP with skype, Block him on Skype, And add a rule in your firewall to ignore all packets from him, You can easily do this in iptables if you were on linux which I assume you aren't. That is TOTALLY possible and will completely work, Also, In your router, You can easily just tell it to ignore ping floods, However, A DoS attack isn't necessarily a ping flood, that is just one means by which it can be achieved.
Also, A ping is a completely useful, And legal, Not necessarily harmful thing. It is just a little thing to ask a computer if it is alive or otherwise connected to said network/accessible. Also, Here are some possibly helpful guides for blocking an IP address on windows, Mac OSX, And linux.
Also, Maybe just get a new skype account and a new account on League of Legends and what ever. Also, Check what information you have visible on Skype and League of Legends, They may be able to see your email address, So you may want to use a new one. You could, Also, Just google yourself to see what comes up, And maybe change some stuff as to not be as easy of a target. Regardless, You don't *need* to do this, Just do the stuff below, You can easily block any IP he uses to attack you. You could use skype to keep getting his IP if he tries to change it, So you can just add it to the list again.
Also, You should be able to check in some log file or what ever or maybe an app what IP is connected to you and sending you stuff, The one that is spamming you a lot is most likely him, So just block it through the following means. Regardless, You can always respond to me here and I will see if I can do anything else to help you. More information would be helpful.
Windows Guides:
http://www.cm3solutions.com/block-ip-address-ip-range-using-windows-firewall/
http://superuser.com/questions/340093/windows-firewall-simple-way-to-block-an-ip-address-that-is-scanning-user-keep
http://www.ehow.com/how_6828963_block-ip-address-windows-firewall.html
Mac OSX Guides:
http://www.dummies.com/how-to/content/how-to-use-mac-os-x-snow-leopards-builtin-firewall.html
https://discussions.apple.com/thread/1860372?start=0&tstart=0 (The last post)
http://tcpblock.wordpress.com/ (This may be useful, but not necessarily.)
http://lisaandroger.com/2009/11/mac-os-x-ipfw-firewall/ (This "Water Roof" application seems to be helpful?)
http://osxdaily.com/2010/04/16/block-and-manage-ip-blacklists-easily-with-peerguardian-for-mac-os-x/ (I guess you could add the IP to the list?)
Linux Guides:
http://www.cyberciti.biz/faq/how-do-i-block-an-ip-on-my-linux-server/
http://www.cyberciti.biz/faq/how-do-i-block-an-ip-on-my-linux-server/
http://jadendreamer.wordpress.com/2013/04/18/linux-tutorial-blocking-ip-addresses-from-your-server/ (the save command here sounds important)
https://www.linode.com/docs/security/firewalls/control-network-traffic-with-iptables (A slightly different command is provided here which may or may not be valuable.)