I don't get Wi-Fi if I step away from router?

I got an iPhone 6 a little over a month ago. I had no problems connecting to Wi-Fi anywhere (school, home etc). As of two weeks ago or so, the Wi-Fi would drop in and out. This wasn't TOO bad or worrisome because I had the same issue on my Android. It got gradually worse, it would stop working for hours on end. Some day were better than others. As of 2 or 3 days ago, it just stopped working completely. So I went to the router and there, perfect Wi-Fi! I walked several feet away and it got weaker and weaker until it stopped by the time I reached my room.
My fine connects and works on Wi-Fi networks just fine anywhere else. We have iPhones, Android's, tablets and a PC that work fine on the home network. I've tried trouble shooting by restarting my phone, turning router on and off, erasing phone network settings, making sure I have latest ios etc.

From the symptoms your describe, then there are three possibilities:

1) Another nearby network is operating on the same or an adjacent channel - log on to the router, go to the wireless settings and try different channels.

2) There's some form of fault in your router.

3) The router is one that has adjustable wireless power levels, and the level has some how been set to a lower level than it was previously. Not all routers have this facility.

Without knowing what router you have, it is difficult to give any more advice.

It sounds like maybe 1) there are too many channels in use nearby, consider a) using an app to show Wi-Fi channels usage b) changing your router to a different channel. 2) you are mixing Wi-Fi types eg having an OLD wep device talking 802.11b/g/n whilst also using a newer 802.11ac device using 5ghz/2.4ghz and your router's/and/or/your new device, is getting a bit miffed with something.Investigate what DUAL Wi-Fi features your router has, consider enabling them(usually a signal called GUEST), and moving old devices over to backward WEP bg/n Wi-Fi.

Hmm, iPhone can't run a Wi-Fi Analyzer program.

One, use 5.4 GHz if at all possible. It has more trouble going through walls and the like so neighbors are less of a problem.
Two, find a way to learn what channels everyone else around you are using. Set your router to use the channels with the weakest signals.
Three, check the antennas. Make sure they are screwed in tight. If your router doesn't have external antennas, get either a mesh router (expensive, $300 to $600 but very good), or one that does 802.11ac and does have this. Point the antennas in different directions.

Assuming the phone works fine as you have trouble at home, not elsewhere.