Of course- in fact, in the Forgotten Realms setting, not having a patron deity means you can only be restored through divine intervention,
miracle spell, or
wish spell. This is because those who served no deity are branded as the Faithless, and their spirits become the walls of Kelemvor's realm.
The major Faerunian deities are (along with alignments):
Mystra (Neutral Good)- Goddess of Magic; maintains (and is) the Weave.
Azuth (Lawful Neutral)- God of arcane spellcasters.
Helm (Lawful Neutral)- God of obedience, duty.
Mielekki (Neutral Good)- Goddess of the forest.
Silvanus (Neutral)- Nature deity.
Oghma (Neutral)- Deity of knowledge.
Lathander (Neutral Good)- God of the sunrise.
Bane (Lawful Evil)- God of tyranny and fear.
Cyric (Chaotic Evil)- God of deceit, murder.
Shar (Neutral Evil)- Goddess of the night; creator of the Shadow Weave.
Corellion Larethian (Chaotic Good)- Chief god of elves; god of poetry, magic, and war.
Gruumsh (Chaotic Evil)- Chief god of orcs.
Moradin (Lawful Good)- Chief god of dwarves; god of craftsmanship.
Yondalla (Lawful Good)- Chief god of halflings.
Bahamut (Lawful Good)- Dragon deity; king of all good dragons.
Tiamat (Lawful Evil)- Dragon deity; ruler of evil dragons.
Kelemvor (Neutral)- God of the dead.
Talos (Chaotic Evil)- God of storms.
Umberlee (Chaotic Evil)- Evil goddess of the sea.
Torm (Lawful Good)- God of righteousness.
Tyr (Lawful Good)- God of justice.
Ilmater (Lawful Good)- God of suffering, altruism.
Malar (Chaotic Evil)- God of beasts.
Garl Glittergold (Neutral Good)- Chief god of gnomes.
Kurtulmak (Lawful Evil)- Chief god of kobolds.
Maglubiyet (Neutral Evil)- Chief god of goblins.
Waukeen (Neutral)- God of merchants (and money).
Chauntea (Neutral Good)- Goddess of the harvest, earth.
Shakandur (Chaotic Neutral, and yes I misspelled it)- God of traveling.
Tymora (Chaotic Good)- Goddess of luck, good fortune.
Beshaba (Chaotic Evil)- Goddess of bad luck, misfortune.
Clerics follow this rule for choosing a deity: They must be within one step of the deity's alignment. However, they CANNOT choose true neutral alignment unless the deity is also true neutral. Some deities have exceptions to this (namely, Mystra and Oghma). Mystra can be used as LN or NG for cleric purposes, as any alignment for a cleric is acceptable for Oghma. Also, for clerics of racial deities, you must be of that race to be a cleric for that deity.
To highlight this, suppose we have a cleric of Lathander, who is neutral good. You can be one step away on either axis- you could be Lawful Good, Neutral Good, or Chaotic Good. In this case, you cannot pick true neutral because of the other rule above. However, if we have a cleric of Corellion Larethian (Chaotic Good), the cleric's alignment can be Neutral Good, Chaotic Good, or Chaotic Neutral.
For non-clerics, technically you can choose any deity regardless of your alignment. However, remember that choosing a patron deity means you have some sort of connection to the ideals it upholds. It is almost impossible for a lawful good person to be a follower of Cyric, the god of deceit and murder.
Also, try to give me an idea of where your character comes from.
EDIT: To determine stats, roll a six-sided dice 4 four times, drop the lowest roll, and reroll any 1s. Do this six times, and assign the scores in any individual stat you like.
What class are you wanting to be, Link?