Ah, you're lucky I chanced upon this topic. Allow me to introduce myself, I am Holy Assassin. The self proclaimed expert on assassins in Oblivion. Build-wise and playstyle-wise atleast. I don't know a whole bunch of trivia though.
Anyway, first thing I'll do is show you my build, and why I like it, and how it works out well for me (and I've heard several other people around here who have appreciated my opinions in the past)
Here what I consider the best assassin build:
Race: Khajiit Vampire - Khajiits have a natural Night Eye lesser power. Something that's incredibly useful for assassins. Any race would work though, so it doesn't matter that much. Vampires have boosted Illusion and an array of benefits that would help assassins greatly. Again it is not essential, simply personal preference. I was willing to put up with the disadvantages, you might not be, but that's okay.
Birthsign: The Theif - This is pretty much the best way to go about it, but you might find others like The Steed to to useful aswell. Even The Lady could be useful.
Class: (Name it whatever you want if you choose custom, but remember; there is nothing wrong with default classes. Look through each one carefully, and see if you like a default one more)
Favored Attributes: Speed, Agility - Agility is a must, I would say, but Speed is debatable. Some might say Endurance would be more useful. I stuck to Speed though, because ofcourse assassins need that both from a practical perspective and an RP perspective.
Major Skills:
Acrobatics - Jumping up into high places as a vantage point is useful for assassins. It's also fun to have. It may be swapped with a different one though, if you wish.
Alchemy - Making good poisons is essential to getting those elusive "one hit kills". Health potions and buffs are useful if ever you come into direct combat where you will most likely sustain heavy damage. Also a very profitable skill, that can also double as a cover for your character. Alchemist by day, murderer by night.
Blade - As an effective character, you will without a doubt run into situations were using bow and arrow is simply impractical. For example if you're jumped by a bandit. A back-up combat skill is necessary.
Marksmanship - I think it's self explanatory why this is useful. Shoot down foes at a distance.
Illusion - The single most important skill for stealth-based characters (besides Sneak maybe). It boasts a wide array of incredibly useful spells, like Nighteye, Light, Frenzy etc..
Sneak - Again, self explanatory. You need to be unseen to be able to effectively "assassinate" someone. Otherwise you're just fighting them. It also offers a damage multiplier which is useful as assassins generally don't have a lot of bulk.
Light Armor - This is another skill that you may swap out if you wish. A true assassin would never actually come into a direct combat situation, but we know that Oblivion tends to thrust you into such situations, so it may serve well as a backup skill.
Skills that you may swap in:
Security - It sounds like an obvious decision to use this as a major, but the game's lockpick system is rather easy to use despite having a low skill level, and after level 10 you get the option to acquire a certain item that makes Security an obsolete skill. Up until then, you might need it though. Again, personal preference here.
Conjuration - If perhaps you need a helping hand every now and then, you might want to throw conjuration in there. You could summon creatures, weapons, and armor to aid you.
Skills that you should avoid using as Majors:
Athletics - Under no circumstances should this skill be in any Major list. Ever. It levels itself anyway no matter what character you're using. A waste of space. It's just as effective as a Minor as it is as a Major.
Mercantile - You are an assassin, not a thief. Don't confuse the two. Adding this would be waste of a slot.
Speechcraft - This one levels incredibly slow, and the speechcraft minigame is boring, and irritating for most. Besides, assassins aren't exactly known to be the most sociable of people. Some may say that it would be needed to avoid suspicion, but to avoid suspicion, one doesn't have to be an expert orator. Simply wear a smile. Infact, people might get suspicious as to why someone might want to be so friendly with everyone all the time...
Okay, I know this whole thing was a huge wall of text, but you asked for it! By no means is this the definitive way to play the game though, remember that. Im also not claiming that this is the best assassin build ever made; just my personal experiences put together. Everyone has their own playstyle and build preferences. I respect that. If however you like my ideas, and wish to know more, and be further educated in the Art of Holy Assassination, simply PM me, asking whatever you want to know and I'll fill you in! If there's any point in the above that you find dubious, or wish to discuss, don't hesitate pointing it out either.