The Telerik control should allow you to show the file name. Your extra textbox seems redundant.
Read the telerik documentation and specifically the html and css section. It says the following:
During and after uploading, a file list is displayed, which shows the progress and status of each uploaded file. Its HTML output is the following:
<ul class="t-upload-files t-reset">
<li class="t-file">
<span class="t-icon t-success">uploaded</span>
<span class="t-filename">some-uploaded-file.jpg</span>
<button class="t-button t-button-icontext t-upload-action" type="button">
<span class="t-icon t-delete"></span>Remove</button>
</li>
<li class="t-file">
<span class="t-icon t-fail">failed</span>
<span class="t-filename">some-failed-file.jpg
<span class="t-progress">
<span style="width: 75%;" class="t-progress-status">
</span></span>
</span>
<button class="t-button t-button-icontext t-upload-action" type="button">
<span class="t-icon t-retry"></span>Retry</button>
</li>
</ul>
If you want to update a textbox of your own, simply use their client api by adding this when creating telerik control
@(Html.Telerik().Upload().Name(BrowseButton)
.ClientEvents(events => events.OnSelect("onSelect"))
and in your javascript
function onSelect(e) {
// Array with information about the uploaded files
$('#yourTextboxId').val(e.files[0].name);
}
Again all the details are in the documentation.