...magus? I mean, I should get the most obvious out of the way.
It is a magic class suited for casting in melee. The general strategy for them on this board is 'get the traits that reduce metamagic costs for one type of spell' and then 'fill up your entire spell list with shocking grasps'.
You can attack with a weapon while casting your shocking grasp, and even transfer the shocking grasp to a weapon to get another hit covered in the lightning.
Overall, this seems like the default for 'shocking grasp' build.
If you're starting at level 1, yeah, magus is the way. If you're starting 'deep into the levels' then blade adept arcanist 6 / inspired blade swashbuckler (or something) 1 / eldritch knight is possible. A blood conduit bloodrager might be possible from level 4.
Pure wizard? I guess you could do it with the spirit binder archetype and your familiar delivering the spell. It's possible to make a wizard who can handle a bit of melee, but their buffing actions are likely to be taken up with polymorphing into something with a bunch of natural attacks and so they'd have less use for shocking grasp.
Edit: an arcane trickster might have some use for it. And since wiz 3 / (anything with sneak attack) 1 / arcane trickster is now possible you could be effective there. An eldritch scoundrel is the less impressive version for those allergic to multiclassing.
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I use shocking grasp a lot for my pure wizard. I recommend the following combo;
-Take the trait Magical Lineage or Wayang Spellhunter, applied to Shocking Grasp to lower the effective level of SG when you apply metamagic to it.
-Take the feat Intensified Spell, apply it to your Shocking Grasps and raise the damage die cap from 5d6 to 10d6 (actual caster level dependant, of course). The trait you took in the previous section keeps the spell in 1st level spell slots so you can prepare more.
As for delivering the spell without putting yourself in melee range? Screw putting your familiar at risk or screw having a familiar at all and go with a bonded item. Use Spectral Hand. It will let you make touch attacks at medium spell range with a +2 to your attack roll, but not suffer any of the usual ranged touch attack nonsense (-4 for shooting into melee, -4 cover) since it's still a *melee* touch attack, just at range. That said, don't dump your Strength.
Also, always be aware of whether your target is wearing metal armor or using a metal weapon. The +3 to attack rolls Shocking Grasp provides in those situations is nothing to scoff at.
If you want Shocking Grasp to be your signature spell for later levels, make sure to invest in a Lesser Elemental Metamagic Rod so you can switch up the electrical damage when needed.
When you get to even higher levels and can learn 4th level spells, take Mnemonic Enhancer. If you have roughly 48 hours in game to prepare for a big battle, fill all your 4th level or higher spell slots with Mnemonic Enhancer on day 1. Immediately before resting for day 2 (battle day), use the Prepare option of ME to give yourself 3 extra levels of spells to prep per casting of ME. Fill them up with Shocking Grasp or whatever, go to sleep, then prep the rest of your spells in the morning with the phantom slots from the night before still in effect for the remainder of ME's 24 hour duration (from the time you cast it).
Using that trick, a 20th level wizard that used all 24 base spell slots from levels 4-9 would be able to prepare an additional 72 levels of spells (no higher than level 3, mind you). Bare in mind, casting those 24 castings of Mnemonic Enhancer would take you 4 hours plus the 1 hour to prep the actual spells into the phantom slots. Add in your 8 hours of rest and hour to prep your normal spell slots so you'd have only 10 hours to use up those 72 phantom levels.
A Ring of Sustenance and or the Arcane Discovery Fast Study would buy you more time to play, of course...
[edit] Last thing. Only a Wizard can pull of this fun little trick. Mnemonic Enhancer is one of the sweetest Wizard-only spells in the game.
Toss those Shocking Grasps as a ranged touch spell. You can only get Shocking Grasp as a Witch with the Elements Patron though.
Blood Conduit Bloodrager can swift-action a Shocking Grasp after punching or tripping someone.
If you really want to work Shocking Grasp into something beyond a middling damage bump, metamagic like Rime Spell or Dazing Spell would be the obvious option, eventually crossed with Persistent Spell to keep the DC fierce. Snapping a Dazing Shocking Grasp on someone is still only a level 3 spell with a metamagic trait, which means a level 6 Blade Arcanist can spellstrike it with Potent Magic adding a +2DC. A level 8 Blade Arcanist (or equivalent Blade Arcanist/ EK) can move up with Dimensional Slide, spellstrike with +2DC Dazing Shocking Grasp, and then spellstrike them again with Quickened Shocking Grasp, which is a little more interesting than the usual Magus business.
Sorcerer can add a fair deal of damage potential through bloodlines, which could work fairly well for an electric Dragon Disciple type thing, or a crossblooded Elemental(Cold)/Draconic(Silver) based character that throws a ~67 damage entangling Empowered Intensified Rime Shocking Grasp by 10.
A Aberrant Bloodline Sorcerer after 3rd level get an extra 5ft reach. So this can help keep you out of combat and deliver SG.
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You might look at the Stormborn Sorceror Bloodline from APG.
Good thematics, will up the SaveDC of Shocking Grasp by 1 as well, and has a lot of other neat zappy-themed abilities.
Finally, if this is something you want to use frequently, I highly recommend Spectral Hand as another spell, so you can do it from safe distances.
You can also look at "Shock Shield" and "Shocking Images" as great spells that will help with a build with this theme as well. Shocking Images are a nice way to deal damage during your opponent's initiative if they do find a way to close to melee with you at higher levels. It will definitely make them think twice about continuing to swing, when they aren't hitting you, but still getting zapped every time they try.
Shocking grasp is the go to spell for a magus, the standard dervish dancing scimitar build with optional bladebound, hexcrafter or kensai archetypes works fine, magical lineage shocking grasp +metamagic.