You must not misrepresent the copyright ownership of the software. Therefore, you cannot remove a copyright notice, or replace it with your name. The original authors still hold the copyright to their work.
When you make changes to the software, the copyright of the software is shared between you and the original authors. You can therefore add a notice for your copyright, for example:
Copyright 2017 Your Name
Copyright 2017 Original Authors
While you should not remove copyright notices that are shown in the user interface, you can change how these notices are displayed. A menu item called “Help > About this software” is a good place in GUIs, so you could move them there.
The Apache License 2.0 has a couple of terms that are relevant to your scenario.
You must keep any copyright notices intact in the source files, i.e. as part of the copyright headers in each source code file (see 4.c). This suggests (but doesn't explicitly state) that you are not required to keep copyright notices intact in non-source forms of the software, e.g. as part of a GUI. I would still advise against removing existing copyright notices if that would create the fraudulent impression that you would be the sole copyright owner.
Adding a copyright notice is explicitly allowed in 4.d: “You may add Your own copyright statement to Your modifications”.
If the original contains a NOTICE file, you must display it in the user interface, “if and wherever such third-party notices normally appear” (from 4.d again). For a GUI, that would probably be an entry in the help menu. Again, adding your own notices is allowed: “You may add Your own attribution notices within Derivative Works that You distribute, alongside or as an addendum to the NOTICE text from the Work” (ibid.).
Don't forget to give your users a copy of the license (see 4.a).