AccountController.php

<?php

namespace App\Http\Controllers;

use Illuminate\Http\Request;
use App\User;
use App\Http\Requests;
use Illuminate\Support\Facades\Auth;

class AccountController extends Controller
{
    /**
     * Display a listing of the resource.
     *
     * @return \Illuminate\Http\Response
     */
    public function index()
    {
        //
        return view('Account.index');
    }

    /**
     * Show the form for creating a new resource.
     *
     * @return \Illuminate\Http\Response
     */
    public function create()
    {
        //
    }

    /**
     * Store a newly created resource in storage.
     *
     * @param  \Illuminate\Http\Request  $request
     * @return \Illuminate\Http\Response
     */
    public function store(Request $request)
    {
        //
    }

    /**
     * Display the specified resource.
     *
     * @param  int  $id
     * @return \Illuminate\Http\Response
     */
    public function show($id)
    {
        //
    }

    /**
     * Show the form for editing the specified resource.
     *
     * @param  int  $id
     * @return \Illuminate\Http\Response
     */
    public function edit($id)
    {
        //
    }

    /**
     * Update the specified resource in storage.
     *
     * @param  \Illuminate\Http\Request  $request
     * @param  int  $id
     * @return \Illuminate\Http\Response
     */
    public function update(Request $request)
    {
        $this->validate($request, [
            'name' => '',
            'email' => '',
            'telephone' => '',
            'job_title' => '',
            'profile_description' => '',
            'education' => '',
        ]);

        User::find(Auth::user()->id)->update([
            'name' => $request->get('name'),
            'email' => $request->get('email'),
            'telephone' => $request->get('telephone'),
            'job_title' => $request->get('job_title'),
            'profile_picture' => $request->get('profile_picture'),
            'profile_description' => $request->get('profile_description'),
            'education' => $request->get('education'),
        ]);

        $userImage = User::find(Auth::user()->id);

        if (!empty($request->file('profile_picture')) && $request->file('profile_picture')->isValid()) {
            $fileName = md5(time() . pathinfo($request->file('profile_picture')->getClientOriginalName(), PATHINFO_FILENAME)) . '.' . pathinfo($request->file('profile_picture')->getClientOriginalName(), PATHINFO_EXTENSION);
            $request->file('profile_picture')->move('profile-pictures', $fileName);

            $userImage->update([
                'profile_picture' => $fileName,
            ]);
        } else {
            $fileName = $request->get('old-image');
        }

        $userImage->update([
            'profile_picture' => $fileName,
        ]);

    }

    /**
     * Remove the specified resource from storage.
     *
     * @param  int  $id
     * @return \Illuminate\Http\Response
     */
    public function destroy($id)
    {
        //
    }

}

AJAX

<script>
                $(document).ready(function(){
                  $('#account-update').on('submit', function(e) {
                    e.preventDefault();
                    var formData = new FormData();

                    // other inputs

                    //append some non-form data also
                    formData.append('name',$('input[name=name]').val()),
                    formData.append('_token',$('input[name=_token]').val()),
                    $.ajax({
                        type: "POST",
                        url: '{{ url('account.update', auth()->user()->id) }}',
                        data: formData,
                        processData: false,
                        contentType: false,
                        dataType: "json",
                        success: function(data, textStatus, jqXHR) {
                           //process data
                        },
                        error: function(data, textStatus, jqXHR) {
                           //process error msg
                        },
                    });
                  });
                });
                </script>

index.blade.php

<form class="form-horizontal" id="account-update" role="form" method="POST" action="{{ url('account.update', auth()->user()->id) }}" enctype="multipart/form-data">
  {{ csrf_field() }}
  <div class="form-group">
    <label for="name" class="col-sm-2 control-label">Name</label>

    <div class="col-sm-10">
      <input name="name" type="text" class="form-control" id="name" placeholder="Name">
    </div>
  </div>
  <div class="form-group">
    <label for="email" class="col-sm-2 control-label">Email</label>

    <div class="col-sm-10">
      <input name="email" type="email" class="form-control" id="email" placeholder="Email">
    </div>
  </div>

  <div class="form-group">
    <label for="telephone" class="col-sm-2 control-label">Contact Number</label>

    <div class="col-sm-10">
      <input name="telephone" type="telephone" class="form-control" id="telephone" placeholder="+(44)754 5635 950">
    </div>
  </div>

  <div class="form-group">
    <label for="job_title" class="col-sm-2 control-label">Job Description</label>

    <div class="col-sm-10">
      <input name="job_title" type="text" class="form-control" id="job_title" placeholder="Web Developer">
    </div>
  </div>

  <div class="form-group">
    <label for="profile_description" class="col-sm-2 control-label">About Me</label>

    <div class="col-sm-10">
      <textarea name="profile_description" class="form-control" id="profile_description" placeholder="Tell us about yourself..."></textarea>
    </div>
  </div>

  <div class="form-group">
    <label for="education" class="col-sm-2 control-label">Education</label>

    <div class="col-sm-10">
      <input name="education" type="text" class="form-control" id="education">
    </div>
  </div>

  <div class="form-group">
    <label for="profile_picture" class="col-sm-2 control-label">Profile Picture</label>

    <div class="col-sm-10">
      <input name="profile_picture" type="file" class="form-control" id="profile_picture">
    </div>
  </div>

  <div class="form-group">
    <div class="col-sm-offset-2 col-sm-10">
      <button type="submit" class="btn send-btn">Submit</button>
    </div>
  </div>
</form>