I frequently play an extreme sort of melee character who I envision as some kind of druidic martial-arts master. The character never uses a gun, rarely uses any sort of weapon apart from a spiked brass knuckles or a power fist when facing difficult challenges, and doesn't care much for armor either--except when fighting Enclave soldiers I like to wear nothing more than a combat leather jacket.
One thing to keep in mind as a hand-to-hand fighter is that if you use a melee weapon you may do more damage, but if you use bare-knuckled punches and kicks you'll almost always get knock-downs and knock-backs on enemies once your skills are developed; these take away the enemy's action points making them much less of a threat (since they can't attack you as much).
Another thing to keep in mind is that as a melee fighter, the bonus move perks will be very important to you. They help you quickly close with the enemy so you can attack, and once you're there they allow you to inflict knock-back attacks on the enemy and still have plenty of AP to move again and continue attacking.
I think both of the builds posted so far vastly underrate endurance. Endurance is the major factor in how many hit points you'll earn, and unlike most other stats can only be boosted by a single point after the game starts (by taking a gain endurance perk). I would go for more endurance, as you'll need more hit points to play a melee fighter. Enemies with guns will simply stand around shooting you as you try to close the distance with them in order to attack, quickly stripping away your health.
Another thing I would change about these builds is starting with 10 agility. If you start with 9 agility you'll have 9 action points, which is plenty until near the end of the game; you can always take a gain agility perk later and bump your AG to 10 to get 10 APs. This would give you both an extra point to put into endurance or charisma.
It's also better to pick one means of attack (either unarmed or melee weapons) and focus on building that skill as high as possible rather than splitting points over two skills (such as using melee weapons at first and switching to unarmed later). Specializing from the start allows you to become that much deadlier with your chosen means of attack in the end.
I always tag the unarmed skill and raise it right away; if you're going to be a melee fighter then the sooner you're able to defend yourself the better, and I don't see much point in waiting around to score a few free bonus points here or there. Apart from the Arroyo and San Francisco trainers, all the other possibilities for bonus points increase your skill regardless of how high it is anyway, so you're actually better off having a high skill when you take these types of training. A five percent bonus is only 4 points if your skill is 80, but would be 8 points if your skill is 160; furthermore the number of skill points required to raise your skill level by normal means from 160 to 168 is 40 free points--therefore using the bonus training is a bargain indeed when your skills are high, and not a very good value if your skills are low. As far as how high to ultimately go with the unarmed skill, I usually aim for 175 by the end of the game.
Early in the game my perk priorities are usually multiple levels of toughness and bonus move. Bonus HtH Attacks is an excellent perk to take as soon as possible, as are Slayer followed by Better Criticals. Some other important perks to keep in mind are Living Anatomy and the Combat Implants operations. Bonus HtH Damage and More Criticals aren't that useful due to the way they work, and I rarely take either of them.
Strength is very important of course for a melee fighter, but there's not a lot of reason to go for a high strength in the character creation screen for several reasons. There are many ways to raise strength for free in the game (power armor, BOS chips, perk), and after you get Slayer your ability to inflict devastating critical hits will be much more important than your base melee damage score (based on strength). Getting Slayer in the first place though, requires a high strength--so you'll either want to have some power armor or some buffout handy to temporarily boost your strength at level-up time once you approach level 24 (the minimum level to get Slayer).
For traits I usually take gifted and small frame, as both give extra stat points and small frame seems like a good choice for a female Chosen One (my usual choice). In my current game for example my starting character was:
S: 6
P: 7
E: 7
C: 6
I: 7
A: 9
L: 6
Traits: Gifted, Small Frame
Tag skills: Unarmed, Speech, Outdoorsman
This is a very well-rounded character who can develop into a fierce melee fighter while not missing out on much at all in terms of quests that have other skill level requirements. The development strategy generally depends on taking a few gain stat perks, getting the BOS computer implants in San Francisco, getting a +2 luck from the Hubologist in NCR, and getting defensive bonuses from combat implants and becoming a prizefighter in New Reno.
I'm near the end of a game with this character (the only things left are the Military Base and the Enclave); here's where the character is currently:
Code:
FALLOUT
VAULT-13 PERSONNEL RECORD
22 November 2246 1104 hours
Name: Yaminda Age: 21 Gender: Female
Level: 44 Exp: 963,266 Next Level: 990,000
::: Statistics :::
Strength: 07 Hit Points: 273/273 Sequence: 07
Perception: 07 Armor Class: 040 Healing Rate: 02
Endurance: 08 Action Points: 10 Critical Chance: 008%
Charisma: 10 Melee Damage: 02 Carry Weight: 140 lbs.
Intelligence: 08 Damage Res.: 084%
Agility: 10 Radiation Res.: 046%
Luck: 08 Poison Res.: 050%
::: Traits :::
Small Frame
Gifted
::: Perks :::
Bonus HtH Attacks
Bonus Move (2)
Toughness (3)
Educated
Better Criticals
Slayer
Dodger
Gecko Skinning
Dermal Impact Armor
Phoenix Armor Implants
Vault City Inoculations
Gain Endurance
Gain Charisma
Gain Agility
Living Anatomy
Vault City Training
Expert Excrement Expeditor
::: Skills :::
Small Guns ..... 101%
Big Guns ....... 010%
Energy Weapons . 025%
Unarmed ........ 170%
Melee Weapons .. 070%
Throwing ....... 030%
First aid ...... 092%
Doctor ......... 106%
Sneak .......... 025%
Lockpick ....... 101%
Steal .......... 020%
Traps .......... 102%
Science ........ 160%
Repair ......... 130%
Speech ......... 145%
Barter ......... 065%
Gambling ....... 030%
Outdoorsman .... 106%
A few things to note about this character snapshot:
--I took one level of Educated very early (my very first perk, in fact) to offset the skill point loss from Gifted; over about 40 level-ups this one level of Educated has given me an extra 80 skill points;
--The normal damage resistance of 84 percent (in other words, nearly invulnerable except to critical hits) was while wearing combat armor; with no armor of any kind normal damage resistance is still 44 percent--which is pretty tough.
--This character was run in an RP 2 game, so charisma was boosted by 2 points during by the BOS bunker operation rather than 1 point as in a normal game; she's also wearing mirrored shades (+1 CHA).
--Apart from six or seven skill points spent, the Small Guns skill of 101 percent was attained solely through the use of books; since I'm not really using this skill, there was no reason to hurry it by actually spending points on it. I only recently raised it from the 90s to 101 because I don't have much else I need to spend skill points on at this point near the end of the game.
--Since I haven't been to the Military Base yet, perception will go up by one point when I go there and get the bonus perception chip for the BOS computer;
--If I gain a few more levels before the end of the game, additional perks I'll take will be Mr. Fixit (as I noted in relation to unarmed earlier, my science and repair skills are high enough now that a ten percent boost is a real bargain), and perhaps an Action Boy or two (combat mobility = victory). Nonetheless the Action Boys aren't nearly as useful to a melee fighter as Bonus Move, which is why I never took them earlier.
--At this point the character is nearly invincible. Wearing combat armor she's able to defeat smaller Enclave patrols by herself, and using a mega power fist she kills each trooper in one or two punches. No other enemies present much of a threat with the possible exception of packs of fire geckos--if one of them knocks her down and the rest breathe fire, it can be a problem. If she's mobile, though, she kills fire geckos in one punch and can take out three or four of them in one round of combat.
Hopefully something in the above will give you some other angles from which to look at building your melee fighter character.