$('.do_uploadFile').click(function() {
  $(this).closest(".upload_section").find(".uploadFile").click();
});
$('.uploadFile').on('change', function() {
  // ... code upload file
});

<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.0.0/jquery.min.js"></script>
<form class="upload_section">
  <input type="button" value="Choose file(s)" class="do_uploadFile">
  <input type="file" name="uploadFile[]" class="uploadFile" multiple="multiple">
</form>

<form class="upload_section">
  <input type="button" value="Choose file(s)" class="do_uploadFile">
  <input type="file" name="uploadFile[]" class="uploadFile" multiple="multiple">
</form>

<form class="upload_section">
  <input type="button" value="Choose file(s)" class="do_uploadFile">
  <input type="file" name="uploadFile[]" class="uploadFile" multiple="multiple">
</form>