Here's a list of the Lamborghini cars and their body types for career races, though these miles can be accumulated in ANY mode. (DLC cars have the DLC pack listed in parenthesis)
GRAND TOURING LEAGUES
Early Grand Touring (PI 600 or lower)
1988 Lamborghini Countach LP5000 QV
Middle Grand Touring (PI 700 or lower)
1997 Lamborghini Diablo SV
Modern Grand Touring (PI 700 or lower)
2011 Lamborghini Gallardo LP570-4 Superleggera
2011 Lamborghini Gallardo LP570-4 Superleggera (Day One)
EXOTIC LEAGUES
Modern Hypercars (PI 900 or lower)
2013 Lamborghini Veneno (Hot Wheels pack)
Ultimate Grand Touring (PI 800 or lower)
2012 Lamborghini Aventador LP700-4
2010 Lamborghini Murcielago LP670-4 SV
2008 Lamborghini Reventon (IGN pack)
VINTAGE LEAGUES
Grand Touring Legends (PI 500 or lower)
1967 Lamborghini Miura P400
This challenge must be earned by August 5, 2014. Hopefully you'll get a high affinity level for Lamborghini, but look towards the Long Beach and Nurburgring DLC achievements to see if you can stack up on anything, too. If you absolutely just want to get done with this as soon as possible, just tune your fastest Lamborghini, go to free play and do 50 laps on the Indy Oval track 4 times. It'll take you 2.5 - 3 hours via this method.
Quick update: Just wanted to drop a little more info. I had yet to complete all 6 possible championships with the Lambos before this challenge appeared. I had completed a few races prior, and I repeated a few races during, but when I completed all 6 championships I had 400 miles exactly (80% done with the challenge).