What's going on with my laptop? Unable to connect to internet?

Hi! I have a Sony VAIO that I have been more than happy with for over a year now. Though recently a malware virus got into my computer. I restarted my computer through Safe mode with networking, downloaded an anti-malware program and was able to get rid of it. While downloading the program in Safe mode I had to use LAN/proxy to connect to the internet. Ever since then I have had ALOT of trouble being on the internet for more than 2 minutes at a time. At the bottom right hand side of the screen (where it shows signal strength) I get a gray square, which I suppose is represents a connection, with a red 'X' through it. I run diagnostics, and at the end of it it tells me that the problem has been fixed, but it hasn't, sending me through an endless circle of frustrating attempts to solve the problem. I have gone to toolbar>options>under the hood>proxy settings>controls>LAN settings and unchecked everything in case it was still trying to connect me through LAN, but it hasn't helped the problem. I use google chrome and when I attempt to look at a website it says "Error 105 (net:: ERR_NAME_NOT_RESOLVED): unable to resolve the server's DNS address" I'm a 19 year old girl and not exactly computer savy, but I was wondering if this is a problem that anyone else has had or is a problem that can be easily fixed? Thank you for any and all help, it is GREATLY appreciated.