I'm new to the game and have the following Mirkwood/Lothlorien bits -

Galadriel
Celeborn
Thranduil
6x Guards of the Galadhrim Court
4X Galadhrim Commanders
6x Galadhrim Warriors

I also have King of the Dead, and 12x Warriors of the Dead which if I'm reading the allies rules correctly I can also use. What should I be looking to get to round the army out, and how many points are tournament games usually played at?

In my opinion I think legolas or tauriel are better than thranduil due to their special rules, and I believe tournaments are 500 or 750 pts im not sure though having never been to one

how many points you want this?

Ultimately I'd like it to be playable in a tournament, which is why I asked about the points values they're usually played. In the more immediate term, I'm interested in what should go in right now. I've no idea what kind of balance I should be striking between heroes and troops, whether I need cavalry, should I be mixing in things from other good armies, and which are the troops I should be avoiding/definitely should be fielding.

If you like cavalry, there certainly is no reason not to field them (all Elven cavalry is decent, although I don't know how the new ones of the Mirkwood variety fare), but at the same time it's not strictly necessary. Gives some tactical options, and the speed is very useful in some scenarios. Unlike others, these can charge through woodland terrain, which may allow you to catch your opponent off-guard. If you get cavalry, try to keep a (preferably mounted) hero near them.

Unless you play an all-hero army (which is probably a bit tricky to start off with), you will want to try and max out all warbands, i.e. a 1:12 ratio of heroes and warriors. So you'll be looking at a good few extra troops, but given the background, those could be Galadhrim (cavalry or infantry), basic Wood Elves or some of the various new Mirkwood figures.
That said, although there isn't much thematic justification, you should be able to get a decent 500pts force with the current collection if allying in the Dead. The latter are worth some 300pts and are a very solid front line, though lacking in Figh value and Might. Complement that with Thranduil (cheapest 3 Might source you have with great supporting spells) and the Galadhrim warriors, and you have something that may be pretty decent. For an entirely Elven army (or to get this to 700pts), add the Stormcaller for another Nature's Wrath caster and give him a warband of Elves, whatever type you choose to purchase.

Thanks