Hi TimJohnst,
dn already linked the tutorial post which is the first place I would start. The same basic principles apply to heavier people as anyone else except you will need to put a special emphasis on more durable materials and constructions and probably on mattresses that have firmer comfort and support layers (firmer materials feel softer for heavier people and firmer support layers are usually important to for good alignment for higher weights). I would especially make sure you read the durability guidelines here and post #4 here about the factors that can affect durability and the useful life of a mattress. Post #2 here has some generic guidelines for different body types and sleeping positions, the first part of post #2 here also has more information about couples that have a larger weight differential and post #14 here has more about the benefits of thicker comfort layers and thicker mattresses (most of these are in the tutorial post but I thought I’d highlight them here as well)
There are also a number of forum posts and threads about mattresses and higher weights and a forum title search on heavy (you can just click this) will bring up many of them. Some examples include post #2 here and post #2 here and this thread
The choice of materials and components is a personal preference but it’s important to make sure you choose a durable version of whatever materials you prefer. Latex (you can read more about the different types in post #6 here), 5 lb or higher memory foam, 2 lb or higher polyfoam, and stronger innersprings with more steel or in the case of lower coil counts with lower gauge springs (around 13 gauge or lower) are all among some of the more durable materials and components that can work well for higher weight ranges. I would make sure that any lower quality foams are less than around an inch or so in a quilting layer because more than that can become the weak link in a mattress … especially for someone that is heavier (unless of course the mattress is for shorter term use or for a guest bedroom where durability isn’t as big an issue).
The biggest benefit of a king size mattress is more room (it gives each person the equivalent of a twin size mattress) but there will also be more space in the middle that isn’t used as often which can lead to some impressions on each side of the mattress and a “hill” in the middle as the center isn’t slept on as much and some of the materials may shift towards the middle (see post #20 here) which are more common in larger sizes. the most important thing is to make sure that you have durable materials in your mattress no matter which size is best for you so that foam softening and impressions doesn’t lead to the premature loss of comfort and support that means the mattress will need to be replaced.
Some of the better options and possibilities I’m aware of in the Albany, NY area are listed in post #4 here.
Hope this helps 
Phoenix