Depending on game mode, of course, you will always want to start with a blessing, boots, and potions.

I normally start with warriors blessing solely because the extra health bonus and health restored could very easily give this God the upper hand against any opponent you might face.

Boots will give you the minor movement speed you might need early gave on top of a slight boost in power that could make early game much easier on the competitive level.

Because this god does not rely on mana for any of his abilities, the two health potions can keep you in the game much longer than a god that needs mana to push.

I start with Ninja Tabi on this god. Due to his already overwhelming power throughout early, mid, and late game I prefer to have the extra attack speed than the power boost. Because Cu Chulainn focuses on his rage building (by attacking and getting attacked), this allows for faster rage build while still providing some extra power.

Next I normally build Gladiators's Shield for the health restored. This alone change keep you in the game and let you be more aggressive than comfortable giving you an early game push advantage. Add the physical protection, cool-down reduction and power boost to make the deal sweeter and this item becomes the perfect addition to Cu Chulainn's build. (I sometime build this item first if the team is all physical or primarily)

The next item built is normally Mystical Mail. I build this solely to add onto the aura power Cu Chulainn has already built into his kit. This extra damage not only does some serious damage while in a fight, but also fills your rage as you go due to it being considered an attack. Another advantage is the fact that this item does a conflicting magic damage compared to Cu Chulainn's physical damage. This alone makes it a very good item in a 1v1 situation. Of course, the health bonus, CCR and physical protections are a huge bonus as well.

After that I build Ancile. The physical power and magical protections that come with a slight cool-down reduction can make for a very good item for a very aggressive god. It boosts the players ability to survive and the stunning shock-wave is very nice if you need to push a mage in a pinch.

Hastened Katana is a fun and unexpected addition to Cu Chulainn's build. This allows the god to effortlessly follow his target, build rage super fast due to the overwhelming attack speed bonus, and gives a movement speed and power boost to make everything sweeter. I highly suggest this.

Finally, I build Crusher. The sole purpose of this item in my build is for the passive DoT that super effective with Cu Chulainn's kit. It's a very useful tool anytime in the game and the attack speed, and power is a great addiction. I also build this as the only source of penetration in this build for a little something against players building against you.

I will build these two items to counter a magic only team or a magic dps character in a 1v1 situation:

I would replace Gladiator's Shield with Shogun's Kusari for starters. The magic protection is agoing to be the primary reason for the switch, but of course the item comes with a decent amount of cool-down reduction and CCR. The other major benefit is the atack speed that comes with this item. This is great for rage building and being aggressive towards a ranged magic user.

I would then replace Mystical Mail with Talisman of Energy. Again, the protections is what we are after, but the health is sweet. The passive on this item is also very crucial when it comes to aggresion in a 1v1 or otherwise chasing. the extra attack speed and movement is the cherry on top with this item.

Because the core build is already very physical protection heavy due to the passives of some of the items, not much more is needed for a 1v1 against a physical god or a primarily physical team:

I would suggest replacing Ancile with Berserker's Shield. This will give you a little boost with the protections and a large boost with attack speed, and it will also give a little kick with your own power. On top of that comes the passive life steal with the item that stacks really well the your attack speed heavy core build. It keeps the player in the game and VERY aggressive in a 1v1.