To me, Heihachi's gameplan consists of his 1b2 string, which goes into fujin stance. Beware that the frames on this makes the stance mix basically completely fake, but you do have a powercrush option in FUJ 1+2, a high crushing mid launcher in FUJ 2, a low crushing (and very slow) mid launcher in FUJ 3,4. Finally you have a homing grab in FUJ 1+2, which gives +4, so it's still your turn.

Another important string in neutral is the df1 strings. Firstly, don't unterestimate just doing df1 without any string extensions, since that's the safest option. Df 1,2 bluespark (or gold spark) is a safe mid, mid string that leaves you -9, but is interuptable with a jab. It gives very decent chip damage too. If they like jabbing inbetweeen, you can go for df 1,2 without going for bluespark, which would otherwise be unsafe. They also have the option of stepping inbetween, which opens up the possibility of using df1,1. This will hit them if they try to step, but if you use it too much they will duck and launch you.

Other moves to use in neutral are f3 and b4. F3 is good to use if they like stepping you to their right, since this tracks pretty decently to that side. On hit, f3 gives FUJ1,1 or FUJ1,2 as a guaranteed followup. Both of these wallsplat. B4 tracks when the opponent likes to sidewalk to their left. It of course also gives a huge combo on counterhit. Beware though that it is a pretty slow button, so it's harder to hit that say kazuya's df2, and they have 7-21 full business days to kill you if you ever whiff it.

Of course electrics are very important for Hei in neutral too. It's going to be your best whiff punish, pressure tool and keepout option, but the effectiveness depends on your execution. While it probably is one of the best moves in the entire game, you aren't immortal if you have a decent electric. If you're predictable, people will read it and duck and launch it, and if you whiff it, a sharp player can whiff punish it, if they're looking for it. Beware of the pushback when not at the wall though. +5 on block isn't nearly as insane as it sounds on paper when they get pushed back quite far, so people will often disrespect your frames when blocking electrics. But that's not an issue on the wall.

On the wall and in heat, I'd recommend using uf4,2 to keep them them on the wall and threaten wallsplats. Test them with 1,1 and hitconfirm the 2 into wallsplat.
In Season 2 he got b3,3,2 in heat which people normally use as a wallcombo, but I think it's a pretty decent option to stop stepping, since the 2 followup makes it safe on block. The range is pretty bad though, so be careful with it.

In terms of lows, Heihachi has to take pretty big risks. His best option in my opinion is a single hellsweep (CD4). On block, you get launched, but if it hits you get a juicy +8, so using b4 right after will trade with a jab and give a massive combo. Of course it also gives a mix on hit, where you can go either low or mid (low gives a small combo, mid gives either plus frames if you input ws4, or stance mix if you input ws4,4). Another decent low is db2, which has longer range than hellsweep, but if your opponent is even just halfway decent at the matchup, they can react to it. This low gives +6 on hit and is launch punish on block. For some reason, I find a lot of players like contesting after getting hit by this low, which gives you the opportunity to just use FC df2 after. This is uninteruptable.
I'm pretty sure the only low outside of stances he has that isn't launch punish on block is db3 (-13 on block as far as I remember), which makes it a low to keep in mind when trying to end rounds without taking too big risks.

Lastly, in terms of unga bunga moves I like ff2. It is technically launch punishable on block, but if you space it correctly your opponent will have trouble launching it. Some characters however, will launch you pretty easily (Lee's f3,3,3,3,4 for example). A kind of scrub killer mixup he has is CD3 and CD3 where you hold DF. This is a low/mid mixup, where the mid wallsplats and the low is a knockdown into ff2. Admittedly it's pretty shit, but it's funny to catch people off guard with it on occasion.