He always seemed like someone who relished pulling the strings from behind more than actually sitting on the Iron Throne. He wanted his daughter Cersei to initially marry Rhaegar, when Mad King ruled Westeros and Tywin was Hand of the King. Later, when Robert Baratheon's rebellion became successful, he turned the Lannister armies against Targaryen and orchestrated the "Sack of King's Landing", and Jaime also ended up killing the Mad King. Now while Jaime killed the king to stop King's Landing from getting destroyed by wildfire, it did play into Tywin's plans, and Jaime could have actually taken the throne for himself and the Lannisters but he didn't, and sat aside until Eddard Stark arrived with his forces.
So, Tywin did the next best thing and married Cersei off to new king Robert. Again, his plans get foiled because Jaime/Cersei can't stop the incest or stay away from each other, and Jaime is still part of the Kingsguard, which forbids him from having a wife or becoming the new Lannister head, which Tywin always wanted from him.
Tywin did hatch plans to assassinate Daenerys discreetly I suppose, but again Jorah Mormont falls in love with Dany and ends up becoming her protector instead, plus nobody could have predicted 3 dragons being born. Tywin's always had good plans but somehow or the other, fate always stops him.
I've personally always been conflicted on him. He's one of my favorite characters but the Sack of King's Landing and Red Wedding make it impossible for me to root for him.