login.js

// Exported in a shared file
const myLoginRoutine = user => new Promise ((resolve, reject) => {
  axios({url: 'auth', data: user, method: 'POST' })
    .then(resp => {
      const token = resp.data.token
      localStorage.setItem('user-token', token) // store the token in localstorage
      resolve(resp)
    })
  .catch(err => {
    localStorage.removeItem('user-token') // if the request fails, remove any possible user token if possible
    reject(err)
  })
})

// in your vue component
import {myLoginRoutine} from 'the-created-shared-file-containing-auth-api-logic'
...
methods: {
 login: function () {
   const { username, password } = this
   myLoginRoutine({ username, password }).then(() => {
     this.$router.push('/')
   })
 }
}
...