You might want to look into getting a TV which can operate on 12 VDC if that is your only reason to install an inverter. Most flat panels of that size have what is commonly called a wall-wart which converts 110VAC to DC on which the TV operates. Simply eliminating the wall wart will serve your purpose.
Battery capacity is a function of how much power you need to operate the boat's electronics. Figure out how much power you may typically consume between re-charge intervals, double that, ensure your alternator can handle that sized battery capacity effectively and you have the answer to your capacity question.
Deep cycle batteries are preferable to automotive start batteries and the so-called "marine" batteries which can be of dubious and needlessly expensive type.