What's a good build to have that does not have Charisma as a dump stat? (7-10)

Truth be told, I've never known how to play a high Charisma character who was not also strong in Intelligence. Intelligence often makes a huge difference in available dialogue options.
(Certainly make it exceed a 3. 9 to start might be better.)
If you intend to end the game without combat, A very high Speech is necessary. In Fallout 1, Science or Medicine at similar values is absolutely necessary as well. A high Science skill is always fun anyway.
If that's not as important to you, don't worry as much about the Intelligence. That's necessary for achieving dialogue options which alter the flow of the game, but not too important if you're just trying to avoid a combat or two and be well-liked.

Since you're already looking at one high stat outside the combat set, you might consider building a character who is simply ill-suited to combat in general. This would mean getting by on wit, charm, and a high Outdoorsmanship skill.
Alternatively, Small Arms in Fallout 1 (and, to a lesser extent, 2) can allow you to snipe from a fair distance to handle most of your problems. That works better without teammates.
I suppose I can see worse ideas that being dashing and pretty, strong (6 at least) and relying on Big Guns. They tend to solve a lot of problems, too.
However, I would recommend Sneak if you want a problem-solving TAG skill to get around your problems. There's other options, but I really feel avoiding unnecessary combats is very important to a charm build, and a lot of people simply aren't in a talking mood! (Note: Outdoorsman is very, very important to this as well.)

If you wish to rely on equipment so as to win unavoidable fights, I recommend either Gambling or Bartering be brought to or near 100%. Gambling in particular you could probably get by with a bit less. I did okay with 70%. Both become unnecessary when your equipment is maxed out, so I wouldn't waste a TAG skill on either, though there might be some role-playing fun in being just that good a barterer. It could be a fun way to learn the maximum number of coins or caps you can carry in one place, and it does mean you get to solve your problems with money. That's oddly satisfying in my experience.

So I'd say:
(Option 1) Raise your intelligence as well. Get your Science or Doctor skills very high. Both is unnecessary in F1, though in F2 Doctor opens a few interesting doors if brought rather high.
(Option 2) Sneak, predict, and solve your problems in roundabout ways, and rely on charm and wit for situations that aren't game-ending in importance. Outdoorsmanship, Sneak, and a solid Perception.
(Option 3) Pick one combat skill to back up enough to solve your problems, but don't build your strategy or whole Perk list around it; just enough to solve intractable problems. Recommend Big Guns and Strength if you go this route. Not efficient, but hilarious and oddly versatile. More practical, of course? Sniping. But just not as hilarious.
(Option 4) Rely on equipment for combat problems, letting that make up for your failings in most stats. Throw your stat points into Charisma, Intelligence, Perception and maybe luck, and equip your (several) buddies very well if you're in Fallout 2. Enjoy having more money than Pluto.

Similar thoughts.
High char + high int lets you talk your way out of many problems.
High int is very useful as you'll have to sacrifice lot of other stats.

I'd keep at least a medium strength (5+), helps a lot when you can carry stuff.
Endurance and agility can be sacrificed, but you'd better let companions lead the way in a fight.

I don't like energy weapons much, but those would make a very nice weapon skill choice.

Fast shot would be a suitable choice, helps to offset having fewer action points and aimed shoits are mostly useful only if you're a good shot to begin with.

Thanks for the pointers everyone. This will be my first playthrough for Fallout 1 but I'm no stranger to the wasteland having played 2, 3, NV and yes even BOS (Side note - don't judge too harshly, but I rather enjoyed BOS. I played it first and was not aware of the controversy of canon. And I needed a fix after coming down off of Champions of Norath). I want a good build where I don't miss out on some good companions and story as I will not likely be playing it twice.

First off, this is Fallout. If you only play through once there is quite a bit you will miss out on.

Second, some additional advice.
Fallout 1 only has 4 companions and only 3 of them are very useful at the end of the game. You want Ian, Tycho and Dogmeat. Katja is not bad per se but you most likley won't encounter her until much latter in the game when she would no longer be useful. Ian loves the SMG, nobody around him loves him having one. He likes to go full auto all the time regardless of who is in front of him. Give Ian your best pistol and plenty of ammo. Tycho likes long guns so give him a rifle or shotgun. Dogmeat is a dog, just let him be.

In Fallout there is a chance to permanently upgrade each SPECIAL by one. Charisma and Luck come through making the right choices in dialogue each with a specific NPC. The rest can be purchased by paying for an operation. Take this into account when building you character. For example you only need 6 Charisma to recruit 3 companions so you can start with 5 and either get the bonus point or pop a mentat. Stength can be raised by 3 by using power armor plus 1 with the operation, so don't put more than 6 in it to start.

The equipment you start with is not the best and it is somewhat dependent on how you created your character. There is some equipment in Vault 13 but they won't let you back in at the very beginning. To get the most use of what they can give you need to go back as early as possible. You must be out of the starting cave for 24 hours IIRC before they let you back in.

The overall Level Cap is 21. This means you will only get 7 perks. Read up on each one and decide which you will want. Adjust your build so that you will be sure to meet the minimum requirements. If you are unsure of what perks to get over others then come back here and ask. I have taken every single perk in the game at one point in time so I can offer my biased opinion.

In Fallout 1 charisma is almost useless. It doesn't affect if and how many NPCs will join you and isn't checked for dialogue either, int determines the dialogue options you get and the speech skill whether you're successful.
There's one check for 7+ charisma if you're male else it doesn't matter. The worst thing which can happen with minimal charisma is that some quest givers don't like you and you have to ask several times.

Stevedog13's suggestion for a character build is quite good if you insist on starting with 7 cha, but you definitely don't have to start with 8 perception, I'd transfer 1 point to strength for more comfortable carrying capacity and weapon use or maybe 1 point into luck for a higher chance of getting good random special encounters. I'd consider tagging energy weapons instead of doctor, especially if you choose "good natured" as trait.
If you play with 2 endurance consider taking fast metabolism.

I second this. Fallout 1 had the best story of the series. I might have to take that back if I find out somehow New Vegas beat it in that regard, but I think that'd be fairly miraculous. Just breaking in to say, I've often found the Charisma upgrade dialogue option simply didn't show up at all. Anybody able to figure out what prompts it?

From what I hear, it may be based on your Speech skill.

I did find at least one dialogue result which was inconsistent with a Charisma of 4. It was one of the first indications I had that, despite similar reactions by one character, the game and its programmers were attentive to gender.

As it is a random encounter I don't have a quick and easy way to test it but I seem to remember that you need to be real friendly and mellow with the guy. The coveration goes something like:

Hey stranger. Where you headed

Nowhere, just wandering

Same as me then. I just wander around selling stuff and singing

What to you sing

Old Irish songs

Do you know this one?

[You spend several hours singing together]



I'd have to open the game and read to code to see if there are and skill checks in there anywhere.

You'd have to, since the dialogue sequence isn't what I lacked, but the mechanics which make it available. Most of my characters get their +1 Charisma from him, but some find they don't know the song.

Alright, here is what I have found so far about raising Charisma through dialogue. You start off by asking:

Whats your Story?

[i]I wander from place to place, making a living with music and a little bit
of tinkering work.[/i]

You then get a choice
What kind of work do you do?
What kind of music do you sing?

Oh, mostly old folk songs and some Celtic music. It's a change of pace.

At this point the game rolls a random(?) number and compares it to your Speech. If you pass the check you get:

Would you happen to know "Na Gheala Mbeadh", as I think it's called?

If not:

I see. What else do you do?


So while it does depend on your Speech skill it seems it more just blind luck. You could get it with a speech skill of 2 or miss it with a speech skill of 110. Also this dialogue only shows up when you first talk to him. In theory you could choose "what else do you do?" and "what kind of music" over and over so that it rerolls the random number until you make the speech check. Also as it is a good character you are talking to you will get better speech modifiers if you have good karma, so that helps too.

I can't remember if this applies to Fallout 1, but in FO2 I was able to get a pretty good game with a CH of 4 and using Mentats / Sunshades / CH Module to make up the difference when needed (i.e. talking my way into Navarro base)

Like MackieStingray said, high IN and speech are the most important--but you may be able to get by on less CH than you'd think.

Ah, but by that logic you can get by with much less intelligence than you might think as well, simply using Mentats every time you want to actually talk to a bunch of important characters at once.
This is not an invalid approach, though it means during the recovery period your character won't be able to form a coherent sentence.