Six feet one inch isn’t a problem for almost any decent kayak–unless you are all legs. Even if your legs are a bit longer than the average six footer, most sea kayaks will accommodate you and some will have adjustable seats. For extra long legs, the depth of the cockpit can be somewhat important when it comes to entering and exiting the boat.
I don’t think you mentioned how much you are willing to spend, but there are some boats that I would suggest if you are considering composites. Current Designs Prana, Stellar standard Intrepid (18’) and the low volume version. If you want to go for about half the price, you might want to look at a Current Designs Sirocco. I know for sure you will fit in that boat and don’t worry about adding weight. Some folks might think that adding weight to a boat to sink it further in the water is the thing to do. I wouldn’t worry about it. I paddle all my boats without adding weight for ballast and they all have over 400 lb. capacity. I am 6’-2" and about 180 lbs.
Most important is to find a boat you like the looks of and will provide the performance you will want. Early on, you should begin to think about whether you want to mess with a ruddered boat. My preference is a skeg, or nothing. You might also want to think about the style of boat that appeals to you. I strongly favor the British style upturned ends.
If it is at all possible, visit at least a few real kayak stores–not sporting goods and big box stores. If you look at used boats, be sure you know what to look for in the way of damaged, warped, or otherwise compromised goods.