There are so many options. For starters, what race do your want to be. You can be any race, but some start out with bonuses that can be helpful, like nighteye, or higher skill/attribute levels that might help an assassin, but any race can be an assassin, and the starting race skills/attributes can easily be overcome.
The next thing I would think about is whether you want to use magic and/or poisons/potions. Do you want to be an assassin that goes invisible to sneak up on victims or to escape, for instance, and do you want to do that with a spell or a potion. If so, then you might want to consider illusion and/or alchemy as major skills. Alchemy levels so quickly that it can be taken as a minor and you can still make pretty good potions/poisons, but if you want to be able to go invisible with a spell, you probably want illusion as a major, since th elowest level invisibility spell in the game requires 50 illusion skill.
Agility actually does not help any assassin skills. So you don't need to worry that much about agility when building an assassin. Agility increases bow damage and adds to total fatigue and reduces stagger in melee combat, but it won't increase your ability to pick pockets open locks or remain undetected while sneaking.
Beyond that, you probably want to take a major skill in whatever type of weapon you plan on using, which for an assassin, I would assume blade, but it can be blunt or you can rely on bows and be a sniper.
As for birthsign, it really depends on personal choice. The mage birthsign is nice if you are going to be a magic using assassin. Another choice for a magic using assassin is the shadow, which gives you a once a day invisibility power, but it also lets you craft spells using invisibility. The lowest level invisibility spell in the game you can buy is a journeyman spell, which you cannot cast until your illusion skill is 50. But if you takle the Shadow borthsign, you can craft a shorter duration invisibility spell as soon as you get access to the arcane university that you can cast with 25 skill in illusion.
There are a lot of options, but that should give you some ideas. You probably want sneak as a major skill since it is so important for assassins. Also security might be a good major, unless you are really good at the lockpicking mini game or are going to rely more on alteration to open locks. If your mark is sleeping in their house, you will need a way to get that lock open so you can go in and do your business.
Light armor is another good major skill for an assassin. You can use light armor as a minor skill, but playing an assassin means you are avoiding getting hit most of the time, so your light armor skill may go up more slowly than for a melee fighter. For that reason, I would consider taking it as a major (unless your assassin does not plan on wearing light armor -- an assassin could just wear cloths).
Good luck and have fun.