There are quite a lot of options for this.

I’m going to start with the simple stuff since you asked a pretty basic question, but I’m also going to expand that into a fair few options that are generally a bit better.

Basic Multiclassing

When your wife hits 5th-level, she can take a level of a spellcasting class, e.g. Wizard. Bam, she now has spells.

You calculate her stats by adding what a Wizard 1 gets to what she already has by virtue of being a Fighter 4. That is, she gets d4+Con HP, +2 to Will, 2+Int skill points, the ability to summon a Familiar, the Scribe Scroll feat, the opportunity to specialize in a school of magic, and a number of 0th- and 1st-level spells each day, including bonus spells for a high Intelligence score.

There is a complication here in that “a wizard begins play with a spellbook [with some spells already in it],” which is sort of ambiguous about what happens when you didn’t start as a Wizard but rather multiclass into the class. I suggest that you give her the spellbook she’d be entitled to shortly after becoming a Wizard, or perhaps as part of the process by which she starts her wizarding career. This is a fine time to get a spellbook. Certainly don’t send her into combat with those stats and no spells, and expect her to be able to fight like a 5th-level character, though.

The rules explicitly and intentionally do not describe exactly how one goes from one level to another, so you are free to describe this however makes the most sense (finally reading up on magic like you promised Ma you would, enrolling in an arcane university, becoming an apprentice, whatever).

Obviously, if she chooses a different spellcasting class (aside from Arcano completo’s Wu Jen, who also uses a spellbook, or the Archivist from Heróis do Horror who uses a “prayerbook”), there is no need to worry about this.

Multiclassing and spellcasting

Basically, spellcasting levels don’t play very nicely with levels in other classes. Where melee characters get things like BAB, which all classes grant, and bonus feats, which you typically qualify for based on your stats rather than your levels, spellcasting is purely level-based. A Fighter 19/Wizard 1 has no better spells than a Wizard 1 does. At 5th-level, when a Wizard 5 has 3rd-level spells, a handful of 0th- and 1st-level spells is not quite as useful. Still useful, no doubt, but not really quite as useful as you’d expect to get at 5th level.

There are some solutions to this. Arcano completo provides the Practiced Spellcaster feat, which provides a bonus to Caster Level up to +4, but is limited by your HD. This means that you can “make up” for the Fighter levels a little bit. You don’t get higher-level or more spells, but at least the spells you have will have the potency they would if cast by a higher-level spellcaster.

Arcane-specific problems: multiclassing and melee fighting

The Sorcerer and Wizard classes have poor BAB and HD, which isn’t good if you want to continue fighting in melee.

Divine spellcasting classes and psionic manifesting classes have much less difficulty here. Neither is affected by Arcane Spell Failure, which is a fairly big problem for a Fighter since armor proficiencies are a major Fighter class feature. Cleric and Druid both have ¾ BAB, which is usually pretty good, and Clerics even get poder divino which gives them full BAB as long as it lasts. On the psionic side, Ardents (Psiônicos completos) and Psychic Warriors get ¾ BAB also, and the War Mind prestige class is great since it gets full BAB and progresses faster than Psychic Warrior.

But you don’t have to give up on arcane spellcasting. There are many options for mitigating Arcane Spell Failure and Prestige Classes can have better martial stats. The Eldritch Knight from the Guia do Mestre has full BAB and a higher HD, though you lose another spellcasting level, which is muito painful, particularly since you have left the realm where Practiced Spellcaster can help you.

Better options include Abjurant Champion (Mago Completo) and, perfect for a Dwarf, the excelente Runesmith (Raças de Pedra). If you have this book, I não podes stress enough how good Runesmith is here for your wife’s character. It allows her to cast her spells while wearing armor, gives her much better stats than Wizard otherwise would, and avoids more lost spellcasting levels. Plus fun stuff like giving other characters “runes” that they can use to cast some of her spells themselves.

I therefore recommend Fighter 4/Wizard 1/Runesmith 5/Abjurant Champion 5. The other 5 levels, if she gets that far, are harder, and I’ll have to revisit this answer if you need suggestions for them. Alternatively, Fighter 4/Cleric 16 (or Fighter 4/Cleric-and-some-Prestige-Class 16) works quite well, as does Fighter 4/Psychic Warrior 16 or Fighter 4/something 1/War Mind 10.

Rules for becoming a multiclass character can be found on pages 59-60 of the Player's Handbook.

The very, very general gist of it is that class features (including spells) are in most cases tracked separately, base numbers (including such vital things as hit points and base saving throws) are added together.

For example, a Fighter taking a level in Sorcerer is going to gain the spellcasting ability and the familiar of a 1st level Sorcerer, 1d4+Constitution modifier hit points, and a +2 to their Will save. If this additional level would place the character's total level at a multiple of 3 (3, 6, 9, 12...) the character gains a feat, and an ability score increase if it is a multiple of four. Again, refer to pages 59-60 of the Player's Handbook.

Know that D&D 3.5 tends to reward specialization in character builds. Frivolous multiclassing can hurt a character's power level to the point where it is difficult to contribute in combat. Making a new character to explore spellcasting may be a more viable option, as has been suggested.

Also know that the "XP for multiclass characters" rules blurb is possibly the most frequently ignored rule of 3.5 (see this brand new discussion). It is usually seen as complicated, counter-intuitive, and unnecessary.

Well, there is two possibilities. Either start a completely new character (which might be a good solution, although the character might need to be made above 1st lvl), or, she can multiclass (thus becoming Ftr 4/Wiz 1).

However, as always, there is a catch. The Fighter and the arcane casters (Wiz, Sorc) aren't really that compatible. The casters usually cannot wear armor, which would limit her function as a Fighter, and aside drom that, the casters are not suited for melee (small HD, low BAB, low Fort and Ref saves, provoking attacks of opportunity while casting, ...) Also, if your wife uses a shield, please note that as a caster, she would require a free hand in order to cast spells with somatic components. Furthermore, she would need the required ability scores to be effective as a caster. (intelligence as Wiz, charisma as a Sorc)

Also, if your wife is looking forward to tossing around powerfull spells, the multiclassing path might not be the best way, because she would only act as a level 1 caster, added to her 4 levels of Fighter. Therefore, her spellcasting ability would be limited, and that might not be exactly what she is looking for.

If you insist on developing a caster from your current fighter build, I suggest you use Cleric. It is a different type of spells, however, clerics can also do some fun things at level one, and do not suffer the armor problems.

Overall however, I strongly suggest building a new character. It might be a pity not to use the one she already has, but even if the Fighter had the required ability scores, which I doubt, it will still not be as caster-like as she would probably like.

Whichever way you choose, have fun playing again!

Rebuild the character from scratch.

This answer goes outside the normal bounds of character advancement, but might be appropriate if you don't mind ignoring the normal advancement rules and want to make a very large change to the character. If you go this route, make sure to justify it with in-game story: she picks up some kind of divine artifact, gains (or is possessed by) the memories of a long-dead wizard, dines heartily at the wrong kind of fairy feast, is blessed (or cursed) by a demigod/lich/magic beastie, etc. I'm making the (possibly excessive) assumption that your wife has suddenly decided to change classes, rather than just tacking on some small amount of spellcasting onto her existing character. (If you want that small, incremental addition of spellcasting, then take a level of Cleric, like the other answers suggest.)

Build a new character, from scratch, that reflects what your wife wants. Pure arcane spellcaster? Go ahead and make a wizard, sorcerer, or whatever else you want. Hybrid caster/fighter? Make a Cleric, a Duskblade, or any other class that allows that. This will let your wife taste the full power of that class at her current level, rather than adding a couple of 0th- and 1st-level spells to her existing character. As you build this new character, try to make it "look" as much like the old character as you can; it's going to be the old character once you're done with all of the number-crunching of character creation. Go ahead and shuffle her characteristics a little; it's harder to be a full divine caster if you don't have a good Wis score.