F1 2013

TL;DR? The achievement will pop once you complete one-shot qualifying in your first race for your 5th team.

To get a contract to race for a new team, you must place (not necessarily win a race) above a rival driver in a best 2 out of 3 races. However, there are only 10 races in this game mode. So you have only 1 screwup race allowed to finish behind your rival, since race 9 (Korea) or 10 (Brazil) has to be on the 5th and final team.

It would be easy to say that you should just choose bottom feeder rivals, but remember that you get their car and team upon contract, so your performance will drop as well in the next competition.

The important thing to remember is that each team has two drivers, so it is easy oversight to simply choose 4 different rivals throughout a Season Challenge. You need four different teams, however, so pay attention when you are choosing, as the decision is final. There are eleven teams in F1 2013 with the following pairs, listed in order of team rank

Infiniti Red Bull Racing
(1) Sebastian Vettel
(2) Mark Webber

Scuderia Ferrari
(1) Fernando Alonso
(2) Felipe Massa

Vodafone McLaren Mercedes
(1) Jenson Button
(2) Sergio Pérez

Lotus F1 Team
(1) Kimi Räikkönen
(2) Romain Grosjean

Mercedes AMG Petronas F1 Team
(1) Nico Rosberg
(2) Lewis Hamilton

Sauber F1 Team
(1) Nico Hülkenberg
(2) Esteban Gutiérrez

Sahara Force India F1 Team
(1) Paul di Resta
(2) Adrian Sutil

Williams F1 Team
(1) Pastor Maldonado
(2) Valtteri Bottas

Scuderia Toro Rosso
(1) Jean-Éric Vergne
(2) Daniel Ricciardo

Caterham F1 Team
(1) Charles Pic
(2) Giedo van der Garde

Marussia F1 Team
(1) Jules Bianchi
(2) Max Chilton

There are 10 races in a Season. Each one needs 2 to defeat a rival. Subsequently sign the Team with their Rivals. And the whole pull through until the end of the Season. Simple Thing.