A mate did this, deep chasing, plastic trunking up the wall - both behind the TV and "somewhere else". Run your cables up the trunking across (in my mate's case in the attic), and down the other. It does need "quite large" trunking to get pre-terminated HDMI cables through - IIRC we used something like 40x25mm trunking at the side, and 25x16 behind the TV, but I'm not sure about that and we did in fact run out of room down to the TV and have to change our plans
As an alternative to using HDMI cables, you could use HDMI-over-Cat5/6 extenders which means threading unterminated Cat5e or Cat6 cables through and then terminating them. Also, there's an option (which my mate was using anyway) of having an external device manage the connections and just have one HDMI cable to the TV - in my mate's case he used a surround sound system with the various HDMI sources (console, DVD, Sky) connected to it.
Of course, whether you can chase that deep depends on what your walls are made of - in my house you'd "struggle" getting that much out of the 'kin hard brick.
As for remotes, you can get remote extenders - the most basic are just a receiver you put somewhere convenient, and one or more transmitters that will repeat the signal inside the cupboard. But make sure there's enough ventilation if you put stuff in a cupboard.
