As bremner suggests you could use AutoCorrect in Word (Tools, AutoCorrect, Replace 0, With Ø) to get the Ø (Alt 0216) character each time 0 is entered. This works in Times New Roman and should work in several other fonts.
Alternatively you could use the Fixedsys font which enters a 0 with a dot in the middle of the 0.
Before doing something like this one has to think it through. Capital O slash - Ø - is not a zero, so it won't work in spreadsheets.
The best solution is to choose suitable fonts. Some have much sharper distinctions between 0 and O than others. Try Georgia or Garamond for example.
You can also edit fonts if you have special needs. Font Creator from High-Logic click here is an excellent and low cost Truetype font editor. Adding a slash to the zero on a few dozen fonts wouldn't take long. One would simply make it into a composite character like â or é.
Changing the keyboard mapping is certainly possible in Word or WordPerfect. To change it system wide, the most flexible solution and easiest to implement is a macro recorder program. I use Perfect Keyboard Pro click here to type special symbols required for Pali like Ā Ī and Ū. One could map the O slash to Control zero, then one can still type a true zero easily.