I'm a big tank player. Playing melee breaks down to observation of the battlefield. It takes practice but you have to learn to spot your openings and you also have to understand that there are some opponents that will not give you any openings if you're alone.
Shyvana is a great melee fighter, however, pre-6 (before level 6) she does not have hard engage/disengage. You rely on Burnout for your fast in and fast out.
For Shyvana I recommend taking Burnout at level 1. Be aware that if you only use it to move, the speed and aoe damage only lasts 3 seconds, but if you land a basic attack while it's up, then it lasts 4 seconds. I use this almost exclusively for wave clear. Clearing waves keeps pressure on your enemy and if they are also melee, they won't want to cs while you are among your minions on fire. It also give you a way to harass: If you run near the enemy champ and hurt them with Burnout, it will not draw minion aggro. Whereas, if you basic attack the enemy, it will draw minion aggro.
As a melee champ, minion aggro is your best friend. You want to force your enemy to take minion aggro if they want to attack you. Stay close to your minions as much as possible. Focus on cs and let the enemy champ do whatever (dodge their skillshots, fall back among your minions if you see them approach to attack, use Burnout to keep them away from cs while taking as much cs as possible yourself. Your best course of action early game is to out-harass and out-last them.
At level 2, I take Flame Breath. With this, you can stand behind your minions and shoot the skillshot damaging the enemy minions and harassing the champ. Flame Breath applies a mark that as long as the sigil is above them, attacks made against them (champs and creeps) deal bonus damage based on max HP.
At level 3 I take Twin Bite. I use this to secure CS when the enemy champ isn't around. It's best use is in conjunction with Flame Breath. Hit the enemy champ from behind your minions with Flame Breath. Activate Burnout to run in quick. Twin bite the enemy champ and back off before minion aggro gets you. You will deal a significant chunk of damage.
Continue to chunk them down in this way. Once you hit 6, just chunk them until they are low and then Dragon Dive them for the kill.
After that it's all mechanics. Make sure to move around A LOT, Bait them out by approaching them and then turning back toward your minions. Hit them once then fall behind your minions to drop the aggro. It's a chess game when you play melee. They move, you respond. You attack, they respond. Try to control the game. Shyvana's only "CC" comes at 6 as her Dragon Dive can push the target. Be aware that no matter how far you push the target, you always land on the far side of them. If you dive toward their tower onto them, you will end up between them and their tower. This can be good, but be careful not to land under their tower. You'll die.
I hope some of this helps you.
