I'm wanting to make a straight board and mace warrior orc with heavy armor. Anyone having any issues not using archery or destruction to down the dragons? This will be my first serious melee only (other than hunting deer) character

You'll only get dragon souls when a dragon attacks you and there is absolutely nothing else in the area.

Lots of people will get killed by dragons because you can't stagger them from afar with a shield unless you're captain america.

Nope. Dragon shouts and companions help tons.

I've done it. Having Lydia along, and high-tailing it to the widest area in the vicinity, is your best bet. Also, bringing enough potions for a regiment is helpful.

I have been playing on my melee character, and every time I have encountered a dragon he will fly around for a bit and then decide to try and land on top of me (next to me on the ground) and I just take them down from there. I am strict warrior, no archery or magic. So you shouldn't have to much of a problem.

Awesome. I don't really use companions so that is out. I'm playing this game for solo stuff not to have someone I have to worry about killing every 10 mins running next to me.

I don't use archery on my paladin and yeah, dragon fights can get rather long if they decide not to land, but eventually they will once in a while. Just make the best of the time they are on the ground, at low HP they are forced to land and stay landed.

And another question...how does one come across hide for leather for smithing without a bow? Are there good areas to go where I may be able to get the deer/elk?

Getting elemental resistance perk or a certain Daedric Artifact help a lot. Board and sword is totally viable but without ranged you can't take the initiative in fights.

Wolves, bears, cats etc. Just wander around, they'll find you.

My Orc has never shot a bow, he does just fine.

Killing dragons in melee is easy. You just have to wait for them to drop down and obligingly let you hack them to death. As long as you have healing potions dragons are easy. (Yes, even the blood and frost dragons.) I would recommend you develop alchemy and smithing for a melee only character, though.