I've had my iPhone 5 and simultaneously Verizon since July 2013. Since that time, I've mostly lived away from home in an area where I have LTE and WiFi provided by my university and I've had no problems. Now that I'm home, however, where we have 3G at best and absolutely nothing most of the time, I'd rather use our WiFi as to not drain my battery/data/etc. The WiFi works perfectly on all our devices (laptops, game consoles, tablets) but not my phone. It rarely makes a connection and when it does, the signal strength is so poor I might as well not even be using it. I've reset the network settings twice and we've even changed our router and nothing has fixed my phone's connection to them. I'm not really sure what else I can do?
Have you tried resetting your network settings on the iPhone? It would be Settings >General>Reset>Reset Network Settings. I had this problem with my iPhone 5 and this seemed to fix it. Also, I'm a Comcast customer. They have these wireless gateways (modem/router in one) that had known issues with iOS devices having poor connections. I had this problem too so swapped it for a standalone modem and bought a standalone router. It doesn't sound like it's a router issue though if it's not happening with any devices other than your iPhone.