Diana has strong poke with her (q) as long as it lands, may be a little hard to angle properly. Her dueling skills are very strong, making her a very good jungler, her clear is also very good. The best start for her is Krugs then you can either go straight for the red buff or get wraiths and secure scuttle. her (w) helps with taking less damage so its a must in any sort of dueling, her ult is a very strong gap closer, but it would be wise to land you (q) before using it, since there is less of a second cooldown if it lands. This build focuses on making Diana tanker than usual to give her the ability to make mistakes, and still do a whole lot of damage.

Diana Is a strong dueler, she has decent poke and can easily nuke squishies late game. However Diana doesn't have the best early game till level 6. Diana can poke down the enemy and cs at the same time with her q, and respond to engages using e-w.

When Jungling I usually start krugs->raptors/wraiths->Scuttle to secure vision->red buff (to allow smite to be up again, maximizing the health u leave the top jg with->blue buff.
Diana's w allows for the best first clear, for it gives diana a shield and does a bit of damage, right after first clear i get her q for the extra damage. Her e can only be used to group up the camp to maximize q damage, but thats about it, it does no damage and it may just be a waste of mana
When in Lane, i start with my q to poke and cs, i lower down multiple minions to get an excuse to poke with my q and get cs. Her passive also allows for good csing.

Diana can easily start team fights, land a q and you can throw ur whole combo, problem is if the enemy expects you to engage. q-r-e-w will group the team right where you are, if you have zhonias hour glass your pretty much have a lissadra like engage your whole team can then follow up and get the kills, you may die if you do not have zhonias or if u you get caught out, it is risky to engage. however if the player your engaging is the AD carry or an AP carry you should be fine as long as they arent near any hard cc tanks.
Lissandra Engage:

Diana doesnt have the very best of ganks, most of the time if you miss your q the gank has already failed. To engage ganks land q-r and once they are barely of the range of your moonfall (e) use e-w, if ur team mates cannot get the kills after a couple autos, use your ultimate to claim the kill, if they are low enough that is. saving ult can be very useful for when they begin to run away with low health and her moonfall (e) is very good to stop people from running away. when ganking try get your team a couple of the kills, but claim one or two to at least make yourself more than useful.