Long story short, yes, you can make a viable melee fighter. You do want to have SOME kind of ranged attack, however. You know, for those times when someone is shooting at you from the second floor balcony and you can't get to the stairs.
For a melee build, I would pin Agility, Reaction, and Intuition as your key stats, though you'll want to have a Body, Strength, and Willpower of at least 3 in each. Agility is what allows you to hit, and all those net hits add directly to damage. It doesn't matter if you can punch a hole in a tank if you can't hit it. Reaction and Intuition are key, not only for being your Initiative score, but also for being your ranged defense pool. This is VERY important, as 'not getting hit' is one of the key points in surviving a gunfight.
Now, you can do a damn fine martial artist build as an unaugmented mundane, but let's face it, you're going to be in a lot of trouble once you move past street-level foes. So let's just assume you're going either with augmentation or magic. Either way, you have two routes to pick from. Because of the costs (nuyen/essence with 'ware, karma/PP with magic), it is hard to do both, or at least do both well. The two routes are Offense and Defense. Armor is no longer capped by Body, so in this case Defense means Not Getting Hit. Either route, you want to have at least a +2d6 to initiative, but going the Defense route means you get the wired reflexes, the Reaction Enhancers, dermal plating, and so on. That'll eat up most of your essence, to say nothing of nuyen. Going Offense means increasing AGI and STR, getting as many bonuses to your primary combat skill as you can, improving accuracy as much as possible, and in general just making sure you harder and more often. This can eat a good bit of nuyen, but is not as essence-dependent when talking ware. On the magic side of things, Adept/Mystic Adept is the way you want to go for a melee-focused character, though some Magicians, especially Possession tradition mages, can do some nasty things in melee. There are plenty of combat-focused powers to choose from, but again you're going to be going with the Offense and Defense routes, because you only have so much karma/PP to spend.
Which way you choose to go depends on the character and your playstyle. Are you going to be the chromed-out razorboy, or the ninja assassin?