I just started playing a melee character a few weeks ago: Rachel B Riddike (kind of obvious who she is inspired by). She started with basically the best club she lay her hands on, then quickly moved to a combat knife (not the best damage, but very quick, so lots of hits), then finally the Shishkebab. I have to say I love the latter, as the flaming blade looks cool, and it is fun to set everything you hit on fire. I did find a mod that removes the motorcycle tank from your back, because I got sick of it blocking my view.
As far as stat-wise, I put more points into Strength, and obviously poured a lot into the melee weapons skill. Hitting is pretty much guaranteed, but the higher your skill, the more damage you do. At first level it was pretty tough, as you get beat on a lot. Even mole rats were a challenge when I was using the police baton from the vault security guards. But once I got the knife (and a few levels), things got easier. The hardest part is getting close. It is a good idea to use stealth a lot to sneak up on people. Or if you are spotted, find something to hide behind and let them come to you. In buildings just duck behind a corner and wait. Once they come up, smack them! You do not need to use v.a.t.s., but I seem to get more criticals when I do. So I usually use when I am making a sneak attack or lying in wait for someone like that. Often it is a one-hit kill.
For a backup, I use heavy weapons, because sometimes they are across the Potomac, on top of a building, etc... The Rock-It Launcher is Rachel's main ranged weapon. There is just something so perverse about killing people with tin cans, that it is perfect for her. Sometimes she uses the missile launcher because it is fun to blow things up, or the minigun, just because it is easier to aim at targets very far away than the Rock-It Launcher.
For perks, I mostly used them to raise my strength and endurance (you take a lot of hits, especially early on). There is one that raises your DR (called toughness I think?). There is also one I found a mod for called http://www.fallout3nexus.com/downloads/file.php?id=1190 that works the same was the Iron Fist perk, only for Melee Weapons instead of Unarmed, adding extra damage to all your attacks.
One of the bonuses to this kind of character is that you save a lot of money, as you can sell all the guns and ammo you come across, and you never have to buy any.