So I'm currently enjoying life as a smashing orc, meaning that I take down pretty much everything before asking. I don't use spells, nor bows, only do I use swords and axes.
However, I must say that I sometimes face severe difficulty. For instance today where I was to solo a boss to retrieve some plans
during a Companions quest line - I had to transform into a werewolf to kick his behind, and even so, I used three attempts to get the job done.
So my question to you: how do you, as a melee, fighting character, choose your weapons of arms? One handed + shield or a strong, damaging 2 handed?

I carry both. I just started a redguard like your orc that uses 2handed and I carry 1 handed and shield to help out if i die first attempt with 2handed .

I miss the days if Oblivion where I could switch between a longsword and shield and a claymore (great sword) because they were both under the same skill (blade). I find the swords in Skyrim to be too short for me, so I mainly use great swords.

I use Volendrung, since its enchant means you can do limitless power attacks

Have you tried dual weapons? Even at my currently low level, enemies seem to get staggered back from my attacks quite often. Sometimes they dodge though, then they step behind me and bash me in the back...but the only battle I've actually lost so far (using dual swords) was when I fell off a tower.

I also had a horrible time once while fighting an Orc, but my character had Rockjoint at the time.

I haven't tested this myself, but I've read posts from players who have said that they feel unstoppable with dual axes. That might suit an Orc better than swords maybe.

EDIT: I just realized you said "smashing orc" - durrrrrrr! sorry

I will let my post stand as a monument to my inability to read.

I apprecieate all your comments, and yours, Katt!
I might try dual wielding for sure.
However, I find the two handed weapon absolutely amazing for damage! yet, the defense may lack a bit too much...

I run a 2 handed smashing Orc, and it's awesome. Once you can dual enchant just put fortify 2 hand on everything it will let you put it on, and fill in the rest with whatever fits your playing style and you're ready to roll. Cna be tough early on, as some magic wielders and falmer can be a pain to get close enough to for you to destroy them, but, once you get it down, it's pretty easy to let beast mode off its leash and run rampant through all of Skyrim.

Explain "Dual enchant", please

When you get your enchanting to level 100 you can get the perk to put 2 enchantments on 1 item. So, for instance my ring has +40% 2 handed damage and +37% fire resistance or something like that. I believe that you can put +2 handed damage on 4 pieces of equipment, so, it really adds up. With my Deadric Battleaxe and all my enchantments I 3 hit Ancient Dragons.

My Warrior Dude dual wields Daedric Swords well, but I used a Daedric Greatsword to 'cheat' a few levels and I did OK. That being said:
1) I play on Adept.
2) I wear Legendary heavy armor (Ebony from 21-40; Daedric from 40+).
3) I put Fortify Stamina Regen and Fortify Healing Rate on my armor body piece.