As a person who plays Shyvana, you have a couple options. Build tons of damage and jump on squishies, bursting them down and usually chasing them down to finish the kill if they manage to escape the initiation.
The problem I run into with this build is that some champs, if they survive the first bit, do a very good job of kiting you from there and pick up the kill instead because you get tunnel vision. Once she uses her ultimate, all she has is her MV speed buff from W.
The second option is going bruiser/tank and building tenacity to counter her major weakness and make sticking to people a lot easier and dealing consistent damage over a longer period of time which is great for drawn out team fights.
Countering that is more difficult, as there are fewer champs capable of kiting the build on her but have more time for team mates to come to their aid in a sticky situation.
Overall, she's pretty well balanced and her win rate is a good reflection of that as well. Her pick rate on the other hand is just a reflection of the meta. She has great clear times and can split push like a baws but only has one knock back in the form of her ult for CC which is the main culprit for her lower pick rate.
If you don't have a comp built to deal with her, she can be oppressive and carry a game. She is very situational when it comes to drafting and building your team or countering.