Stevedog13: Are you wanting to be a melee fighter or unarmed fighter? Both are very viable and I have finished the game many times with each, but they require slightly different character building strategies. My personal favorite is the sneaky unarmed fighter, sort of ninja like, who always travels alone. However the switch from unarmed to melee or from solo to group play is better made at the time of character creation.
Dartpaw86: Unarmed. Well, fisticuffs really. I mean, there are weapons like brass knuckles in Fallout 3 but I am not sure if they were in the original games.
The original Fallout has Brass Knuckles, Spiked Knuckles and the Power Fist. So if you just want to run around punching everyone, it's got you covered.
Okay, the first thing you want to do is to flesh out a archetype for your character, are you building a simple meat head who just likes hitting things or a stealthy guerrilla warrior who destroys enemies with bare hands. Is you character the leader of a group, a loaner, a smooth talker or maybe a technical genius who can learn to do anything. Not only will this help you in how you role play the character but it will also help guide character creation and development. Some traits and perks may be more obvious with one character type over another. I'm not going to give you exact numbers and stats I think you should be using, that would take away half the fun in playing the game. I would like to share a few things I have learned over my many playthroughs with unarmed characters.
First a quick rundown of the SPECIAL attributes. There is a way to permanently raise each attribute by one point during the course of the game, so if there is a stat that you really want to be at 10, you might consider raising it to 9 at creation and waiting to see if you can raise it later in the game. There are also drugs and other items that can raise attributes temporarily, this can be useful if you are trying to get a perk but don't have one of the minimum requirements.
Strength - A melee fighter might want this as each point of strength above 5 will increase melee and unarmed damage by 1 point. Keep in mind that increases to damage only raise the max damage numbers, so if a weapon states that it will do 10-20 damage that means you will do an average of 15 points of damage per attack. If you get a +2 attack modifier then the range becomes 10-22 for an average attack of 16. So more strength does help but not by a lot, which is something to keep in mind when your rationing attribute points. If you do want to max it out a 10 you should start it out at 6 since you can get +3 from power armor and +1 from other means.
Perception - Not too important for melee as it's biggest use is long range accuracy. It is useful in general combat, however, as perception also determines the combat sequence, which is basically how the game determines who's turn is next. I wouldn't recommend dropping it below 3 or 4. Of course you may want to raise perception above that as it is also a requirement for some perks that I happen to find very useful.
Endurance - This controls how many hit points you gain at level up. The formula is half your endurance, rounded down, plus 2. This means Endurance works best when it is at an even number. A popular tactic is to set it at an odd number to start with so that when you get the free +1 it actually makes a tangible difference. For my ranged characters I drop this to 4, but since you are going to be bringing fists to a lot of gun fights it would not be a bad idea to raise this. Starting it at 9 and waiting for the extra point is not a bad idea at all, but it's not strictly necessary. A starting number of 7 is perfectly acceptable for a unarmed fighter.
Charisma - Has no purpose in combat and very little use outside of it. This is my favorite dump stat so I always drop it to 1.If you do want to role play a more charismatic character it would be better to pump up your Speech skill than Charisma.
Intelligence - If you want a lot of skill points you should raise this to 9 or 10 right at the beginning. If for some reason you don't ever want to use more than 2-3 skills then you can set it to something lower. Be aware that taking your intelligence below 4 means your character won't have the brain power to engage in most conversations.
Agility - This stat determines the number of action points you get. The higher this stat is, the easier combat will be. Even though it can be raised it by 1 point during the course of gameplay, I like to start it out as 10 on all my combat focused character builds.
Luck - Useful for critical hits, and since you won't have the raw fire power to do a lot of damage most of your fights will depend on getting in some critical hits.
Traits You get two and they stay with you for the whole game. I'm not going to cover them all, just the few I feel are focused on an unarmed character.
Gifted - I pretty much use this trait in all my builds. It gives you a +1 for each SPECIAL attribute but lowers all your skills and gives you less skill points on level up. Having higher starting attributes is an incredible advantage, having fewer skill points hurts a bit, but can be offset by pumping up Intelligence. Lower starting skills aren't too big a deal as there are only a few you will really be using anyway.
Bruiser - Gives you +2 to Strength but -2 Action Points. It sounds good on paper, but Action Points are much more important than Strength for this character. Heavy Handed - Higher melee damage but lower critical hit damage. More damage is okay, but if you are counting on getting lots of critical hits, this trait will hurt more than help.
Finesse - Lower normal damage but better critical hit chance. This is actually pretty great in the early game but if you get the Slayer perk this becomes pretty useless.
One Hander - Gives you a boost with one handed weapons but penalizes you with two handed weapons. Brass Knuckles and Power Fists are considered one handed weapons, which means there is no down side for an unarmed character.
Small Frame - a +1 to Agility but lowers your carrying weight. Since you won't need to lug around heavy weapons and ammo this is not a bad choice at all.
Good Natured - Lowers all your combat skills but raises some of Speach, Barter First Aid and Doctor skills. You were only going to use one combat skill anyways so the downside is not that bad, and the skill that get raised are actually fairly useful throughout the game.