iPhone 6s charging and shaking

l received my iPhone 6S a couple of days ago, and while I'm really happy with it, when it arrived, the cable didn't fit into the wall plug that was provided - I wondered if there was something I was missing that all Apple customers somehow know about, some kind of wall plug activation (I know, dumb, right?) but the usb just doesn't go all the way into the plug, so I can't use it for charging.

This didn't particularly worry me, as I have a few usb wall plugs floating around, only I recently realised that when my phone is plugged into this charger, the screen has a tendency to 'shake' when it's charging and when I'm scrolling - I was initially very worried about this, you can imagine how pleased I was to find out that the problem disappeared when not plugged in! I did some quick searching and found a similar problem where someone suggested a connection with a non-genuine cable, and I was wondering if this might be due to the fact I'm not using an Apple wall plug? If so, could it perhaps be somehow damaging to my phone to be connected to this charger? I don't think it's very likely, but I'm desperate to look after this phone! Should I just invest in an Apple wall charger? (In which case, what a rip off, Apple).

Many thanks in advance!