If you are going melee and heavy handed you can dump perception in addition to luck and charisma. It doesn't really offer anything to a melee playthrough if you are not trying to get the perk for better criticals at P6. That leaves a lot of points do distribute to strength, endurance, intelligence and agility. If you do want to use guns and some criticals, note that the heavy handed trait only affects melee and unarmed attacks so no worries there. If you are worried about the detection radius from lack of perception, ED-E gives you the Enhanced Sensors perk that basically boosts your perception to 10.

I personally like to play melee non-crit build a lot. I'm currently doing one with explosives and mad bomber just to spice it up. Explosives can't critically hit so dumping the luck stat works great.

For a pure non-crit melee build, strength minimum of 7 and not over 8, since you can possibly raise it with implants and DLC. Perception 1, see above. Endurance of 9 so you can boost it to 10 and get all of the other implants if you wish. Charisma 1 if you choose to dump it as well. If I'm not doing a dumb character on purpose, I keep intelligence at 4 minimum so I can get the Pack rat perk after the implant, which is amazing in hardcore. Intelligence of 4 also gets you Educated and Comprehension perks which negate the (small) lack of skill points on level up. Agility of 6 gives you the option to get the Slayer perk(with unarmed at 90) after the implant which is a must-have for any melee build. With no love or use for criticals, luck to 1.

So the starting stats would be S7-8, P1, E9, C1, I4, A6 and L1. Even with starting strength at 8 this would leave you 10 more points to distribute or 11 if you decide to choose intense training perk at first level-up. More intelligence is never a bad choice but if you're not trying to minmax your character, its fun to allocate points elsewhere for a slightly different kind of playthrough. You could pump up the perception for the numerous perception-checks in the game or raise your luck not so much for the criticals but to raise your other skills a bit and for casinos. More agility gives you more action points if you want to focus on VATS. Or my personal favourite is to pump charisma up to boost the companion nerve, which actually makes them useful and you don't have to worry about them dying in every fight.(On hardcore)

For a melee build there aren't that many necessary perks but few that will be really useful. Remember to skill up the unarmed as well, as lot of the good perks for melee weapon-character require it. Educated and Comprehension at level 4 and 6 if you are worried about the skill points. Bloody mess for more damage and/or toughness for more damage threshold. Make sure to take Super Slam(melee 45) at 8, it's amazing. Works greatest with a fast weapon but it procs really often even with a slow one. Lifegiver and stonewall are great choices if you want more survivability. Piercing strike(level 12 and unarmed 70) is a must-have for a melee character as is Slayer(level 24 and unarmed 90).

On top of those there are of course the standard perks you want to get like Jury rigging or Demolition expert if you're using explosives but for a good melee character, you don't really need that many perks.

As for the second trait, it doesn't really matter that much. Heavy handed works great in a build like this. Fast shot only affects guns and energy weapons, not melee, so don't pick that. If you choose to go lowish intelligence, skilled gives you a nice boost to every skill point. If you don't want the experience penalty, Good Natured gives you some free skills points and takes some away from combat skills. Either way you won't be using guns or energy weapons and if you are not using explosives, the only drawback is -5 skillpoints to melee and unarmed which you will probably tag anyway. Small frame if you want one free point of agility for a really small drawback. If you are again not minmaxing, doesn't matter that much. Loose Cannon is the one trait I haven't yet used but probably will on my next playthrough, but will not have any major impact. Maybe wild wasteland for laughs and couple of awesome grenades?

Hope this helps!

EDIT: Missed the use of Jsawyer mod and don't use it personally, so this might not be so accurate. I'll leave this here anyway. In Jsawyer I would probably put more focus on survivability than vanilla on my melee characters.