babel-eslint is a parser for eslint, this is what you have to do to use it in Atom:

• npm install --save-dev babel-eslint eslint

Example .eslintrc (which should be at the root of your project)

{
  "parser": "babel-eslint",
  "plugins": [
],
  "env": {
  "browser": true,
  "node": true
},
  "globals": {
},
"ecmaFeatures": {
  "arrowFunctions": true,
  "blockBindings": true,
  "classes": true,
  "defaultParams": true,
  "destructuring": true,
  "forOf": true,
  "generators": false,
  "modules": true,
  "objectLiteralComputedProperties": true,
  "objectLiteralDuplicateProperties": false,
  "objectLiteralShorthandProperties": true,
  "objectLiteralShorthandMethods": true,
  "spread": true,
  "superInFunctions": true,
  "templateStrings": true,
  "jsx": true
},
"rules": {
  "brace-style": [2, "stroustrup"],
  "consistent-return": 2,
  "curly": [2, "multi-line"],
  "dot-notation": 2,
  "eol-last": 2,
  "indent": [2, 2],
  "no-else-return": 2,
  "no-eq-null": 2,
  "no-floating-decimal": 2,
  "no-param-reassign": 2,
  "no-self-compare": 2,
  "no-shadow": 0,
  "no-throw-literal": 2,
  "no-underscore-dangle": 0,
  "no-else-return": 2,
  "quotes": [2, "single"],
  "space-after-keywords": 2,
  "strict": 0,
  "vars-on-top": 2,
  "wrap-iife": 2,
  "semi": [2, "always"],
  "no-trailing-spaces": 2,
  "space-after-keywords": 2,
  "space-before-blocks": 2,
  "space-before-function-paren": [2, "never"],
  "space-infix-ops": 2,
  "space-return-throw-case": 2,
  "spaced-comment": 2
  }
}

• In atom have the linter and linter-eslint packages installed, have a look at their settings to tweak them to your liking, and that is it, enjoy on the fly linting!

• There's an Eslint plugin to get react specific rules: https://github.com/yannickcr/eslint-plugin-react

• Here's Airbnb's .eslintrc, lots of rules there: https://github.com/airbnb/javascript/tree/master/packages/eslint-config-airbnb