Allow me to hit you with a wall of text, with the hope that some of it might help.
Most unarmed characters I make start out with the Gifted and Small Frame traits. Free stat points are always good, and what do you truly need to carry other than a big hammer?
For skills, you seem set on tagging Melee and Unarmed, which leaves you one free skill. Doctor can give you access to potential defensive upgrades later, as well as overcoming issues you might have with breaking your arms while punching away. Lockpick would give you a quick advantage against people with guns. Fact: Someone locked behind a door cannot come out and shoot you while you kill his friend. You can cut your enemies in half and punch them in small groups with lockpick.
Don't overstack Strength, as your damage will come mostly from critical hits which are based on your skill %. An unarmed character can slink by with 2 STR and rely on his skill. But to use most melee weapons there will be a minimum STR requirement that forces you to waste points here. I would advise putting no more than the minimum for the weapon of your choice (Hammer: 6, Knives: 2, Spears 4).
I have known people to tank their perception score when making unarmed character, but I find a decent score helps in several ways. First it lets you start farther back during random encounters - which at first may seem to be a bad thing - but ultimately you'll be running away from half the encounters anyway, so why not get a head start. It also allows you early access to the Awareness perk, which is essential for playing the game the first time. It will display every bad guy's HP and ammo - a huge plus for people who haven't run through the game a hundred times. Take 5 for the awareness perk, 6 for good measure and opening quests.
Endurance is one of those stats people debate over. Some people are okay with saving and reloading their games ten times an hour. Others play without ever reloading. This is pretty much proportional to how much you want to invest into hit points. Since you are unarmed, options for increasing your natural damage reduction will appear. This makes hit points slightly less advantageous.
As for Charisma. You are the ugly bruiser of the wastes. If you start off with the Gifted trait, you have 2 CHA and should be able to recruit one NPC. If you want your entire team to go "no guns" you can grab Sulik from Klamath. If you want a sniper backing you up, Vic and Cassidy are both available in subsequent towns. Items and other options for raising your CHA will appear in the game if you feel the need to recruit more party members.
A decent intelligence score will allow you to activate most of the quests via dialogue - though finding several side quests are based charisma, perception, and speech. Six intelligence to cover most of your dialogue bases, and then any left over points should go here with the plan that they will become skill % later on.
Max agility, as you are going to have to be running towards your opponents more often than not - and you will have to punch them many times to get similar damage from guns.
Luck. You're unarmed, you'll need all the luck you can get, but you probably will not have points to spare in it.
Close quarters combat characters have a nice advantage early in the game - rats, geckos, and junkies go down quickly to a well trained fist. By the mid-game, however, they encounter considerable trouble as guns get the opportunity to shoot twice for more damage before your character can even close the gap to start punching. Choosing your battles carefully can help with that, as can smart armor selection. By the end game, there are energy-based unarmed/melee weapons that even the playing field, and it becomes much more viable. Power armor also allows one to walk up to lesser foes with impunity and punch them in the groin.
You may have to fiddle around with many tweaks to your character before you have any real success getting him out of the early game. If you have a certain play style in mind, it will be a matter of trying out a build and seeing if it works with your play style. If it is dying before it gets out of a few towns, it may be time to start over.
Per's guide also has an excellent
character creation section, but it approaches it from a completionist point of view - written assuming the player will not miss any of the many "off the path" opportunities to raise stats and skills throughout the game. As a first time player, unless you play sticking to a walk through, you may miss a lot of these - so it is better to be safe in character creation than to assume you might stumble an opportunity in game.
