1. You may need to choose a font that contains the special characters you want. With most recent laptops and operating systems, the fonts commonly loaded by default are "enhanced Unicode" character sets that will have common ones, but the user of the other laptop may have set an older "ANSI only" default. Some OS include the ability to search for an alternate font that contains the char you code, but older ones may only be able to use the default.
2. For the Alt-#### method you've been using, the numbers generally must be typed on the Number Pad. The method is normally called the Alt-NumPad method. Lots of laptops don't have a separate NumPad but use an "embedded" one in which a set of keys within the alpha keyboard can be used as a NumPad. You may have "turn on" the NumPad each time you need a special character (and turn it off to continue). The quick keys needed to toggle the NumPad function on/off vary from one laptop to another. As commented by Newport Boy, the Fn key may be all that's needed, as this is a common (not quite universal) toggle for NumPad on laptops.
3. All Windows versions include a "Character Map" that you should be able to find at Start|Programs|Accessories{System Tools. You can Select the chars you want, Copy them, and paste in a document from which you can copy them one at a time to where you want them. In some versions, when you "click" on a char, the ANSI (decimal)or Unicode (Hex) code will be shown on the lower bar, so you may be able to use this method to find codes you want (see number 5. below).
4. If you use Word, there are a number of "shortcuts," mostly as described by Boho just above. Most consist of pressing Ctl while typing the "accent" and then the character to be accented. Word "autoreplaces" with the accented character when you hit the next key. Some of these shortcuts may work in a few other programs, but they're not necessarily "universal."
5. In most versions of Word still in use, if you type the hex code for a Unicode character, and immediately hit Alt-X, the character appears. The Alt-X is a toggle, so if you put the cursor immediately to the right of a character and hit Alt-X it will show you the hex code you'd need to type to get that char with this method, or you can "read the code" from CharMap for most of them as described above at number 3.
Using "|" to show the cursor position, typing 00C6| and Alt-X should give you Æ. Putting your cursor to show Æ| and hitting Alt-X should show you 00C6. So far as I know, this only works in WORD, but it may have been picked up in some other programs.
6. It may help to "load" a language that uses the chars you want. Recent Windows versions can have multiple languages loaded (at startup) with any one of them selected as the default. With the methods above, you shouldn't have to do this, but it's an available thing that sometimes enhances other features.