First of all DNS leaks are less dramatic as they sound and are even less dramatic in the reality, although when following the concept of Anonymity it still is a good and just a logical desire to fix them! How? Simply use public DNS nameservers.

1. The checking part: http://www.dnsleaktest.com/

Now when connected to the VPN and if the test reveals your ISP IP, or basically any IP that sounds familar to you/or your Original Country (you should be able to recognise them based on the hostname and location) it is most likely that your DNS is leaking.

2. The fixing part (easiest method for Windows OS) -> more info in this article: here

Go to "Start" -> "Control Panel" -> "Network and Sharing Center" -> "Change Adapter Settings" -> right click the Ethernet and/or WIFI Adapter (not the TAP 32 Adapter!!) -> "Preferences" -> "TCP IPv4" -> "Use the following DNS server addresses"

Use here as example googles or openDNS nameservers..

Google’s Public DNS Servers:
primary: 8.8.8.8
secondary: 8.8.4.4

OpenDNS’s Public DNS Servers:
primary: 208.67.222.222
secondary: 208.67.220.220

Some more Public DNS nameservers even sorted by Country (makes using a russian VPN + DNS simple): here
 

Additionally: if your Router Admin area supports (the most do) adding custom DNS, do it in your Router setup as well!!
 

Thats it, check the test link again and see if your customised DNS IPs show up, if yes, the DNS leak is fixed.