Q:

Why does [ALT+224] return Ó instead of alpha?

I'm on WIN 7. I found that windows is using ANSI, not ASCII. So when I type ALT + 224 i get Ó, instead of α. To get α I have to manually copy it from the windows character map every time I want to use it. As you can imagine, this gets very tedious when I am trying to type in Attic Greek. Is there no combination to enter α? If not, is there some solution to this?  

A:

As you have already discovered, the characters resulting from character codes between 0 and 255 depend entirely on the encoding that is used.
Windows doesn't use neither extended ASCII nor ANSI (usually Windows-1252); it actually depends on the application.
For example, Alt + (2, 2, 4) gives on my machine:

α in Notepad and on the command prompt.

à in Google Chrome's omnibox, but α in its console and this very text area.

In Notepad++, a with ANSI, α with UTF-8.

For a more consistent behavior, just use Unicode character codes:
The key combination Alt + (9, 4, 5) – or Alt + (+, 3, B, 1) if you set the registry key HKEY_CURRENT_USER\Control Panel\Input Method\EnableHexNumpad to 1 – should result in a α in every application that supports that character.
Sadly, that isn't the case:

The decimal char code results in ▒ in IE's address bar, while the hexadecimal one just beeps.

The decimal char code results in ▒ in Notepad++ with ANSI and ¦ with UTF-8.

The hexadecimal char code results in a in Notepad++ with ANSI and α with UTF-8.

Summary

Set HKEY_CURRENT_USER\Control Panel\Input Method\EnableHexNumpad to 1.

Use Alt + (9, 4, 5) or Alt + (+, 3, B, 1) in applications with full Unicode support.

Fall back to trial and error in applications that lack full Unicode support.

