When you ping out to to the web, do the ping spikes still occur in the same manor that they occur when pinging your gateway? If they don't then the ping spikes shown are simply the router failing to respond to the pings right away because it's "busy doing other things."
One thing you could test would be disabling all network adapters except for your wifi adapter, and thing running, ping localhost -t, or ping 127.0.0.1 -t, or ping 192.168.2.* -t (Whatever the local IP of the computer on the wifi network adapter is.) which would ping the computer itself through the network adapter (I'm pretty sure.). If you still got the lag spikes there, then it could be the wireless adapter driver causing some kind of overhead issue.
Another thing to test would be changing the channel bandwidth on the 5Ghz channel to 20Mhz. Yes, this will limit the speed at which you can connect to the router on the 5Ghz channel, but it's worth trying. Some equipment fails when trying to connect with 40/80Mhz extension channels. (Not sure if you'll have this option on an N600 router as the 5Ghz channel may already be limited to 20Mhz. You'll probably see an option for 20Mhz, or an option for 40Mhz.)
Try connecting on the 2.4Ghz frequency just to see if you get the lag spikes on that frequency. If it occurs on the 5Ghz, and not the 2.4, then there could be some hardware incompatibilities, or some hardware failure. Try connecting to a friends wifi, or perhaps your phone hotspot, and see if the lag spikes occur on just your belkin router, or if they occur when connected to other wifi networks.
It could be worth going down to a store with a reasonable return policy, like Staples, or Best Buy, where you could buy a wireless receiver, plug it into the computer and see if you have the same issue with another receiver.
Do you have any other devices connected to the 5Ghz? If so, see if the ping results are the same for them.
I've spent nearly the last 2 years working as over-the-phone technical support for 5-10 ISPs (Taking residential & business customer calls), and if their's two things I've learned, one is that turning something off and on again solves ~20% of all issues and two is that wireless is extremely difficult to troubleshoot properly.