Welcome Nathan, you've got a bit of work ahead of you .
Has the bike sat around being unused by the previous owner for an extended time, or was it still in frequent use ?
Remove the fuel cap, is there a red liner on the inside of the tank, or do you see rust ?
Pull the fuel line off of the fuel tap, or remove the nut that has the fuel line attached to it, there are two different fuel taps that could be installed, turn the tap on and see if you get fuel like Rob has said .
Next would be removing the carb float bowls and see whats 'living' in there .
The ignition coil as mentioned by Rob and Monte is a prime suspect, the original coil is black and gray in color and should be one of your first replacement parts, with you living in a damp coastal area it will give you problems, if it isn't the problem already .
Do the spark plug leads have a metal cap at the spark plug end, if so, they need to be replaced, they are most likely 27-28 years old, again in a damp environment old leads will give you problems .