In an attempt to solve, rather than rebuild this computer I have joined your wonderful website.
Firstly I must explain that I have followed similar threads on your site and acted on their advice before joining you and reading your pinned New Member threads. Hence I have tried to use some of the tools already mentioned including ComboFix (which wouldn't run). Please excuse my leaping into hot oil before looking.
With that in mind can I explain what is going on with this laptop and what has been done so far.
It is a Dell Inspiron N5110 running Windows 7 Home Premium, Service Pack 1. 4.00Gb RAM. It had been protected using Microsoft Security Essentials which has been uninstalled at some point whilst investigating this problem.
I was initially asked to investigate why the machine had stopped browsing the internet, saying that it was 'Unable to connect'. Opening a command prompt I could see that it was connected to the wireless network and had IP addresses etc. It could also successfully ping www.google.com, ibm.com, bbc.co.uk etc.
I connected to the router using a LAN cable just in case it was a wireless driver issue. Still unable to browse but can ping.
Ran iexplore -extoff from a command prompt to turn off the extensions. Still unable to browse.
Downloaded the Firefox 37.0.1 installer using another computer, copied it using a USB key and installed it. Still unable to browse.
Downloaded ADWCleaner, copied it over and installed it. it was unable to access the update server. Ran a scan and cleaned all the malware it found. Still unable to browse.
Downloaded MalwareBytes installer mbam-setup-184.108.40.2068, copied it over, installed and launched it. It was unable to access the update server. Ran a scan, which found more malware but still unable to browse afterwards.
Interestingly subsequent relaunches of MalwareBytes fail. It asks for System run permissions, then doesn't show the GUI interface. Though the process is running in the background.
Browsed this site and looked for similar problems, following several suggestions including gmer, tdsskiller, mbar and Combofix which didn't appear to run. The system is still unable to browse.
Took the hard drive out. This laptop has to be fully disassembled to do this!!! (Dell! I ask you?) Then ran a scan using the latest ESET Smart Security. This took 9 hours and found 243 infections, 236 of which it automatically cleaned. I haven't yet fully investigated the log to identify what wasn't cleaned.
Rebuilt the machine and it still won't browse.
Contacted you. Whilst writing this I have just re-installed Malwarebytes and ran a scan. It has detected another 45 objects.
I would really appreciate a bit of guidance as I would rather not have to rebuild the machine, in the interests of getting to the bottom of it.