I think an offline name change is preferable to an offline skin change.
Basic idea
If the server you are running is online-mode=false, there are a couple of tricks to get your name changed. If you can add a skin to a registered (free) account and search for the name change method (which are many on this site alone and very useful) you might be able to match and get something there.
As an aside, offline mode can be achieved easily if you use MC Patcher - an awesome way to load mods and easy way to get your minecraft.jar all set up without opening and closing it manually. When you get it set and 'Patch' it, you can click test minecraft and auto runs in offline. From there it connects to offline servers just fine.
Anyways this method would seem likely to work on multiple machines (most likely LAN only) but try to get that name change, work it to an account registered and change the char.png. Hopefully this will work out, maybe would be good to not log into the second minecraft.exe with a verified account to avoid having it share the same skin.
Ways to get the name changed
Information found from here:
How do I change player name in minecraft multiplayer in offline mode in Linux?
PC
@SET /P IGN=IGN=
@java -Xms512m -Xmx512m -cp "%APPDATA%/.minecraft/bin/*" - Djava.library.path="%APPDATA%/.minecraft/bin/natives" net.minecraft.client.Minecraft %IGN%***
(by EdGruberman)
Mac
Copy + Paste to new applescript, compile, save as application, launch:
property user_name : "Player"
display dialog "Enter A User Name" default answer "" buttons {"Cancel", "Continue"} default button 2
set the user_name to text returned of the result
set UsrApp to (path to current user folder)
set UsrApp_unx to POSIX path of UsrApp
set MineScript to "java -cp " & UsrApp_unx & "Library/Application\\ Support/minecraft/bin/minecraft.jar:" & UsrApp_unx & "Library/Application\\ Support/minecraft/bin/lwjgl.jar:" & UsrApp_unx & "Library/Application\\ Support/minecraft/bin/lwjgl_util.jar:" & UsrApp_unx & "Library/Application\\ Support/minecraft/bin/jinput.jar: -Djava.library.path=" & UsrApp_unx & "Library/Application\\ Support/minecraft/bin/natives -Xmx1024M -Xms512M net.minecraft.client.Minecraft '" & user_name & "'"
do shell script MineScript
(by Nashroe)
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