To clarify for this achievement, simply beating each race in the season isn't enough for the achievement; you have to get 5 stars on each event too.
Season One includes 5 of 6 event types, only skipping over Gate Drift which is introduced from Season Two.
You can unlock later seasons before fully completing Season One, based on your total # of stars earned.
As for advice on completing the season, the best thing to do is to play through a few events and go for stars, and save the money you make until you can afford the Renault Clio for 45k credits. This isn't the fastest D class car available, but it is required for the last event of the season, so it makes sense to focus on tuning that car as you work through the season, as it quickly becomes much better than the cars you are facing in the first season.
Once you unlock later seasons, the quickest way to earn credits I have found is to replay the latest 'Elimination' event that you can, and aim to come first consistently. I'm currently working through Season 4, and winning the Elimination event there gives me 2500-3000k credits per victory, depending on what medals I get during the race.
Before that point, oddly the very first race of the first season gives the biggest 'win' payout, of 999 credits each time plus whatever bonuses you get, and it's ridiculously easy.