If you want to stick with the ES5 way of handling Promises, you could use the following approach:

const start = console.log(Date.now())

// Create a instantly resolved promise
for (let i = 0, p = Promise.resolve(); i < 1000; i++) {
  // append new promise to the chain
  p = p.then(() => request()
       .then(() => console.log(Date.now() - start));
}

If you can use ES6 methods, you could write this using the async/await pattern like so:

const asyncLoop = async () => {
  const start = console.log(Date.now())

  for (let i = 0; i < 1000; i++) {
    const data = await request();
    console.log(Date.now() - start);
  }
}

asyncLoop();

The chance that you really want to make the requests one after the other is actually pretty small though, so in case you want to make the request simultaneously, but do something after all of them resolved, you can use Promise.all(...)

const start = console.log(Date.now())
const requests = [request1, request2, ...];

Promise.all(requests)
       .then(() => console.log(Date.now() - start));

You can use the async-await pattern here. You can achieve this by changing the code

async function iteration() {
    let start = console.log(Date.now())
    for (let i = 0; i < 1000; i++) {
        const data = await httpRequest()
        console.log(Date.now() - start)
    }

}

async function httpRequest() {
    return new Promise((resolve, reject) => {
        request.then(data => {
            //Do anything you want to do with data
            resolve(data)
        }).catch(error => {
            console.error(`Error in request`)
            reject(error)
        })
    })
}

Explanation:

1. I have moved the request code in a separate function httpRequest and this function will return promise on success
2. I called httpRequest in for loop using await keyword, now loop will wait till the request is completed
3. I have also wrapped for loop code in function since you can only use await inside an async function