In Telerik RadGrid you can enable filtering for template columns automatically - just set AllowFilteringByColumn = true for the entire grid and you should be done. Also make sure that the AllowFiltering property of the template column is set to true (this is the default value). On this online demo the template column in question is TemplateColumn City. Keep in mind that you will need to set the DataField property for the column to a valid data field from the datasource. This ensures that the control properly retrieves the data on which the filtering pattern will be applied.
Alternatively, you can extend the default GridTemplateColumn class (part of the Telerik RadGrid object model) and add a textbox control to it. Thus you will be able to filter the GridTemplateColumn values in exactly the same way as with other Telerik RadGrid column types. See the code under the ASPX/C#/VB.NET tags for further reference.

And here are some additional helpful resources:
Filter list options
Grid known filter functions
Filtering with template column
Filtering -> How-to chapter

Finally, for some swift integration examples with Telerik RadComboBox ("Google suggest" filtering or "Load on demand" grid in combobox) you can visit the links below:
"Google suggest" filtering
"Load on demand" grid in combobox