I'm also going to upgrade a weapon to +5 Chaos if I can manage it
Don't Chaos weapons scale with humanity?
If that's the case then I'd be careful about that, as dying would screw you over magnificently. Lightning/fire might be a better option for a SL 1 run, as they do consistent damage.
Which weapon should I upgrade? I have a +5 Raw Reinforced Club (picked it up as +1 Raw from a Gravelord), a +4 Lightning, +10 normal and three +5 normal. I'm planning on getting a set of +5 clubs in each upgrade tree, other than magic. I was thinking a +5 Chaos Battle Axe, since speed is the key in this fight.
The battle axe is a joke. I recommend the Baldur side sword, which you can get as a drop from the non-rapier Baldur knights in the Undead Chapel. It has a quick moveset, as well as thrusting and slashing attacks. Great versatility. It scales with dexterity, but you can make it lightning/fire and avoid that entirely.
If you're just going to upgrade the weapons you currently have, get that Lightning +4 to +5.
The only problem I've had compared to my knight this far are Darkwraith enemies with their combos that are impossible to get out of once you're caught
Yeah those are nasty. But if you're an endurance-focused knight, you should be able to roll/block all of their attacks fairly easily and then retaliate. Just watch out for their Dark Hand attack.
and the Four Kings. I've learned how to deal with Darkwraiths so now the only problem with them is patience.
If you are still on the Four Kings, the easiest way to beat them is to just suit up in Havel's Armor, get the Ring of Steel Protection on, put the Grass Crest Shield on your back for added stamina regen, and use your other ring slot for whatever you want. If you have an Estus Flask +2 at least (which you should at this point) and at least 10 uses of it, you can just tank them. Make sure you have a high DPS though, 'cause you don't want to be fighting more than one at a time.