It's quite possible even without rushing to KL, even though you might have to be a bit gamey.
First you destroy both Lannister armies, which is quite easily done by Robb's initial force. After that you move your army to friendly territory and disband it. Make sure you have as good relations to the Tullys as possible (send gifts and give honorary titles). Now raise all levies and use the northern ones to defeat the Ironmen and secure the north. In the Riverlands you should get an army of at least 30k, which you have to make sure does not get stomped by the Tyrells until you can send your northern levies south. When you unite your forces you should be able to secure victory through battles alone. Of course the satisfaction of sacking King's Landing is nice too though.
It's better to capture all the Lannisters in Casterly Rock and King's Landing, it lets you put them to the sword which means they wont bother you ever again. I even managed to matrilineally marry a Lannister I captured (Tommen iirc) to Arya, which paid off since he not only inherited and consolidated control over Casterly Rock, but formed the Kingdom of the Rock and slapped down the Martell controlled Iron Throne (married patrilineally to Shireen and the Baratheon line died, ffs) which meant I was very safe up North. Don't assault King's Landing though, just ensure no armies come at you and siege until you have Joffrey, Tommen, Jaime and Cersei. Be careful executing them all when one is still officially king though, I think it'll cause Stannis, Renly, Shireen or one of Robert's bastards to inherit, and they may well hate you with a passion if you didn't intentionally install them. Worst case scenario is that Daenerys or Aegon will auto-inherit and they will grind you into dust with the Golden Company, Targaryen loyalists and dragons.