Competitions run on a timed schedule, so you need to wait until a competition is live in order to take part. Before you can do this you will need to register for the competition.
To join a competition open the menu, then press RB to move over to the 'Events' tab.
This tab will have three sections -
• Schedule - this lists all scheduled competitions that are upcoming.
• Competitions - This provides a calendar view of all upcoming challenges and the area they will take place in.
• Tournaments - This is where you can sign up for the in-game tournaments, which you will need for another achievement.
From the 'Schedule' section take a look through the upcoming competitions and see if you are eligible to join any. They are restricted by level (both min and max) , so early on you may find you're not able to join many (or even any at all). I got lucky when I checked as there was a min level 3 competition due to start within two minutes, but right now they are all out of my level range.
Press X to apply for the competition and then wait for the competition time. You'll get a prompt on screen when it's starting. Travel to the relevant area - also make sure you have the correct fishing license and requisite gear before taking part.
Then, complete 1,000 of them... you don't need to win (for this particular achievement), only participate.
* Note * People will be seen who earned this very early without having completed 1,000 competitions and before many other achievements. For about a day after launch this achievement was unlocking after completing only one competition. A hotfix was applied a day later, so this no longer appears to be earnable early.