With an ultimate magus, it’s more important to keep yourself as close as possible to a single-classed spellcaster on one side than it is to keep the two even.
Thus you want to circumvent as much as possible ultimate magus’s attempts to progress your lower-level class.
Because at 1st, 4th, and 7th level, ultimate magus advances the class with lower caster level (your choice in the case of a tie), this has a tendency to even out your two classes. If you avoid that, and continue to progress in the more advanced class, you will have more of your highest-level spell slots, which are also higher level than they would be.
Practiced Spellcaster is the solution to taking manual control over which class ultimate magus progresses. Practiced Spellcaster increases a class’s caster level (but no other facet of its spellcasting) by up to 4 (capped at the spellcaster’s HD, so the feat is only beneficial to multiclassed spellcasters). Normally, caster level is nice but not as crucial as things like spell slots or higher spell levels. Ultimate magus switches this up, though.
If a Wizard 4/Sorcerer 1 takes Practiced Spellcaster for sorcerer, his weaker class, his sorcerer caster level is 5th. This puts it higher than his wizard spellcasting’s 4th caster level, so first level of ultimate magus advances wizard. This is despite the fact that the ultimate magus’s wizard spellcasting is, for most purposes, more advanced than his sorcerer spellcasting.
Now you have some control over your advancement, and that allows you to maximize your spell slots.
At 4th level, both classes are even, so you should choose wizard. At 7th, barring finding some other bonus to sorcerer caster level, though, you have to accept some sorcerer progression, which is a shame but not the end of the world. You’ll end up “missing” a total of two wizard spellcasting levels (compared to a single-classed wizard), but gain a total of nine sorcerer levels, plus of course all the rest of the ultimate magus class features.
Otherwise, you need to improve the abilities that grant you bonus spells. Consider the beguiler (Player's Handbook II) instead of sorcerer: you'll use Intelligence for both classes.
Versatile Spellcaster is another feat that will make your slots more flexible.
And pearls of power and rings of wizardry can help by letting you reuse slots. Memento magicka are like pearls of power for spontaneous spellcasters but they are 50% more expensive, which is badly overpriced.
You can use the Beast Strike feat (Dragon Magazine 355) to add "your claw or slam damage" to the damage you deal with your Unarmed Strikes or to the damage you deal in a grapple (the "damage your opponent" option, I presume).
You may find the Black Blood Cultist prestige class (Champions of Ruin) interesting. Its 8th level ability allows you to deal the damage of all your natural weapons whenever you succeed on a grapple check. The class has a handbook.
Also, this question may be of use to you, especially the part of @KRyan's answer that has to do with combining natural attacks and Flurry of Blows into the same full attack routine.
I do not think that a way to add your unarmed strike damage to the damage of your other natural weapons exists.
Best Answer
Ultimate magus (Complete Mage and also online here) can advance both wizard and assassin spellcasting, as it requires both arcane spellcasting prepared from a spellbook, à la wizard (or wu jen), and spontaneous arcane spellcasting, which the assassin has by virtue of the fact that, “An assassin casts spells just as a bard does.”
You need at least 2nd-level spells on the prepared side and at least 1st-level spells on the spontaneous, so a 3rd-level wizard/1st-level assassin has the requisite spells (though you'll need more levels because of skill requirements of both assassin and ultimate magus).
The ultimate magus advances both classes on 7/10 levels, and only one class on the other three levels (at 1st, 4th, and 7th). In each case, the class that is advanced is the one with the lower caster level. Since wizard spellcasting is better than assassin spellcasting, it is probably best to try to boost your assassin caster level above your wizard caster level, so you can benefit from full spellcasting progression for your wizard side. Practiced Spellcaster (Complete Arcane) helps a lot with this, but ends up being not enough without additionally possessing the krau sigil.
Ideal Case: Illumian 5th-level wizard/1st-level assassin/10th-level ultimate magus
You want to enter the prestige class ASAP, you want to do it with minimal loss of wizard spellcasting, and you want to advance wizard spellcasting at every ultimate magus level, particularly since you’ll already have the highest-level assassin spells even if you miss three like that.
First, you’ll need Hide and Move Silently as class skills, at least for your wizard levels and ideally for the ultimate magus levels as well. You may also want Disguise in-class, but it’s not a big deal since it caps at 4 ranks required, which you can do cross-class if necessary.
Move Silently is annoyingly difficult; Flexible Mind from Dragon vol. 326 is the only good way to get it as a class skill. Barring that, you could be a viletooth lizardfolk (Dragon Magic) but with LA +1 it might be better to just dip something (though having it be always a class skill is useful). Alternatively, an elf (preferably gray elf or fire elf for the Intellignce bonus) could take Aereni Focus (Move Silently) from Player’s Guide to Eberron to get it always in-class as well as a +3 bonus to it (also counts as Skill Focus if that matters to you).
Hide has two easy options, at least: Unearthed Arcana wizard variants can get it for you at level one, at the low cost of your Scribe Scroll feat. You could either take a Fighter feat, and use it for Martial Study (any Shadow Hand maneuver), or if you are an illusionist, you could take the shadow shaper ACF to just get it directly (and add your Int to Hide at 5th level, which is pretty cool). I prefer the Martial Study route as it sets you up for Assassin’s Stance and also gets Hide in-class for ultimate magus levels. Martial Study and Stance are from Tome of Battle.
I will note that I find the lack of a feat that grants both skills in-class kind of strange; they’re certainly thematically linked and plenty of feats do that sort of thing. An Apprentice feat could reasonably combine the two (Apprentice [ninja] maybe? Hell, Apprentice [assassin]), but none of the examples from Player’s Handbook II actually does so.
If you want Disguise in-class, City Slicker from Races of Destiny is a pretty easy choice. Silverbrow Humans (Dragon Magic) also get it.
Anyway, have 8 ranks in Hide, Move Silently, and Spellcraft by wizard 5th, along with the 4 ranks you need in Disguise and Knowledge (arcana). This requires at least 5 skills points per level (4 if you have Disguise in-class), but even with 2+Int skill points a Wizard should have at least 6, particularly a gray elf who gets an Int boost. Which is good because of course you also want Concentration.
Then at 6th level, take assassin 1st and Practiced Spellcaster [assassin] from Complete Arcane to boost your assassin caster level to 5th, matching your wizard caster level. At 7th, it’s your 1st level of ultimate magus, and your caster levels between wizard and assassin are equal, so you can advance wizard. At this point, you have a choice. You could leave well enough alone, accept that you’ll miss a second Wizard level at ultimate magus 4th, and thereby gain 9th-level assassin spellcasting instead of 8th-level.
Or you could be an illumian with the krau sigil, which provides a +2 caster level bonus quite similar to Practiced Spellcaster.
So my conclusion is an illumian 5th-level wizard/1st-level assassin/10th-level ultimate magus, with 15th-level wizard spellcasting and 8th-level assassin spellcasting.