Before I begin, answer these few questions to yourself (and probably in the comments too).
Is Twitter really important for you or your brand?
Is your competitor really levelling his or her game on Twitter?
Are you finding it tough to catch up?
If your answer is YES then go ahead with the content.
1. Finding Competitor
So the first step for you is to find your competitor(if you don't already know). Also, you need to understand the fact that you should understand the fact that oranges aren't meant to be compared with apples. Like if you are Rolls Royce then Audi isn't your competitor, even though they both manufacture luxury cars.
Coming back to the point, how will you find your competitor? You could simply find that with a Twitter search. Simply type your domain like "Content Marketing" and see who gets the most traction.
2. Tracking & Finding What Works
Here comes the tricky part, how to track their best performing tweets? Now to solve this you need to use an external service, it's called "Tweetastic App" (disclaimer: I'm one of its makers).
So first go ahead and create a free account and connect into your Twitter. There you would find something called "Twinesis".
In the input box enter the username of your competitor. Once you press the explore button you'll see 400 top tweets in free plan(up to 3200 in the paid plan).
By default, it would show the most liked tweets in descending order and that's all we need. There have a look at what types of tweets worked for them, also in the meantime keep using your Twitter account actively.
Now you have already tracked your competitor and it's time to leverage the analyzed data to your own growth.