NASCAR Heat 3

This will come naturally as you play through the game. Online, Race Now, Split Screen, Championship, and Career wins all count towards the achievement; Challenges do not.

To make this as easy as possible, change all of the settings to Easy with Damage and Flags off. To save time, you will want to select Quick Qualify. This will make starting races quicker, and the better you qualify, the easier it will be to win. On Easy, even starting in last will net you a win on most tracks.

I highly recommend you start with the Career mode, as this will allow you to work on many achievements at the same time. You must begin in the lowest series, the Xtreme Dirt Tour, and as you complete seasons and impress teams (using Hot Seat races), you can then move up into the Camping World Truck Series, the Nationwide Series, and finally, the Monster Energy Cup Series.

You can then also decide, at the end of each season, which series you want to compete in. For example, after the first season, you have the option to continue in the Xtreme Dirt Tour AND race in the Camping World Truck Series, or do one or the other. Depending on how many series you are competing in, at one time, will determine which season you actually get the achievement. At most, you could compete in all four and get this in an early season.

While going for this achievement, you are guaranteed to get the following achievements:

Checkered Flag - Won a race
Perfect 10 - Won 10 races
All I Do Is Win - Won 25 races
No Longer Rusty - Won 56 races
Not Intimidated - Won 77 races
Over the Rainbow - Won 94 races

And have the option, if doing Career mode and the other game modes, to get these achievements:

Online Racer - Completed a Multiplayer race
Online Veteran - Completed 10 Multiplayer races
Online Victory - Won a Multiplayer race
Online Domination - Won 10 Multiplayer races
Head to Head Winner - Won a Split Screen race
Couch Champion - Was the Split Screen champion after 4 races
Dirt King - Won a race at every dirt track
Super Speedster - Won a race at every super speedway
Short and Sweet - Won a race at every short track
Best at all the Rest - Won a race at every speedway
Road Warrior - Won a race at every road course
Tour the Tracks - Raced at every track
So it Begins - Raced in the Xtreme Dirt Tour on a team
Getting Noticed - Earned a spot in the Xtreme Dirt Tour playoffs
Dirty Winner - Won an Xtreme Dirt Tour championship
Ridin' Dirty - Completed an Xtreme Dirt Tour season
First of Many - Won the Xtreme Dirt Tour playoffs
Moving On Up - Raced in the NASCAR Camping World Truck series on a team
Getting Somewhere - Earned a spot in the NASCAR Camping World Truck Series playoffs
Truck'n - Complete a NASCAR Camping World Truck Series season
Big Step Up - Won the NASCAR Camping World Truck Series playoffs
V is for Victory - Won a NASCAR Camping World Truck Series championship
Almost There - Racing in the NASCAR Xfinity Series on a team
Time to Shine - Earned a spot in the NASCAR Xfinity Series playoffs
Journeyman - Completed a full NASCAR Xfinity Series season
Who's Next? - Won the NASCAR Xfinity Series playoffs
Unstoppable - Won a NASCAR Xfinity Series championship
A Seat at the Table - Raced in the Monster Energy NASCAR Cup Series on a team
Playing with the Big Boys - Earned a spot in the Monster Energy NASCAR Cup Series playoffs
Best of the Best - Won the Monster Energy NASCAR Cup Series playoffs
The Peak - Won a Monster Energy NASCAR Cup Series championship
No More Yellow Stripes - Completed a full Monster Energy NASCAR Cup Series Season
Established Competitor - Won a race in Career
Just Getting Started - Completed one career season
Seasoned Driver - Completed 5 career seasons
Grizzled Veteran - Completed 10 career seasons
Welcome to the Team - Hired your first employee
Full Team - Hired 12 employees in any single series
Back to School - Trained your first employee
Head of the Class - Maxed out a single stat on an employee
Employee of the Year - Maxed out all stats on one employee (any series)
Wonderful Toys - Maxed out a single department
State of the Art Facilities - Maxed out all departments (any series)
Car Collector - Owned 6 cars in any single series
Rolling in the Dough - Earned 50 million dollars

Which will, at maximum, net you 52 out of the 61 total achievements!

Winning all career series required for achievements, completing challenges and multiplayer should net you 100+ wins.
You can finish remaining rest in Quick race, Dirt Tour series is shortest, as for track I recommend Jefferson if you strugle with winning without qual if you are confident about your skills use Drebin instead.
You can just use 'Race again', don't need to go to menu after each race to count it but from time to time 'finish event' to check your progress because achievement pops only in main menu.