In order to determine which factions might be most suitable to this achievement, I compared a couple of starting stats for each faction. For each faction I had the difficulty bars centered, which I believe is Hard. (Although I don't think it makes any difference to these particular stats)

For the comparison I looked at the following things:
1) The short victory objectives
For this one I gave each faction an ABC ranking depending on how quickly a victory could be achieved. Because a shorter campaign, means less time to screw up.
2) Number of characters
This shows how many (adult male) characters a faction has a the start. Since the fewer characters a faction has, the easier it is to manage their influence.
3) Influence range
For this one I wrote down the lowest and highest influence any character has at the start of the campaign. In hindsight, I should obviously have used the faction leader's influence for the high value, because that's what the achievement is about.
I, however, didn't do that, and to be honest I just can't be bothered to go back and check again. So you'll just have to make do with the total range.
4) Number of estates
I also checked the total number of estates each faction has at the start, as they are the easiest way to influence influence (see the section Influencing Influence for more details on this).

The results are listed in this table:

Looking at the Victory objectives I feel Dyflin, Gwined and Sudreyar are the all suitable factions to obtain this achievement, as their short Kingdom victories are quite manageable. This doesn't mean the other factions are all terrible, just that victory will take longer with them.

Looking at the remaining stats for these 3 factions, I felt Gwined was the best choice for me. This type of achievement isn't exactly up my alley. So I thought: an easily obtainable victory, only 4 characters to manage, and an influence-range of 3 is probably the way to go.

This is only my personal recommendation, so feel free to pick any faction that tickles your fancy.

There are several things that can affect a characters influence. Most are random, a few can be deliberate, or a combination of those two. I've listed some of them here, although I'm most likely forgetting a few. If I did, feel free to let me know in the comments.

1. Estates:
This is the most obvious way to influence a character's influence. If you grant a character an estate it will raise their influence by +1. Similarly if you take the estate away, they will lose 1 influence. However, when managing your estates it is also important to look at the other effects.

Agricultural estates
Granting agricultural estates might raise or lower your faction's food production, unit replenishment rate, or army supplies. Which it is depends on the characters traits, and might change over time (green squares in the image below).

Religious estates
Granting religious estates to a character affects faction allegiance. This is a relatively complex stat but in essence it determines whether your people are loyal to you. The blue icon is good, the red is bad. If a character raises usurper allegiance (red icon), the usurper faction will gradually get bigger in the local province. This will lower public order, and might eventually lead to civil war.
Therefore you want to keep faction loyalty as high as possible, so I would advise only giving religious estates to characters that will improve faction loyalty.
Note: Unlike Agricultural estates, religious estates will not raise your faction leader's influence (purple squares in the image below).


2. Governor:
Making someone governor will raise their influence by +1. Removing them as governor will also remove that effect. However, it will also lower their loyalty by -1.

3. Marriage:
A character's wife will have a specific trait which will apply specific stats to their husband. Among all the various traits are several that also add or remove influence. If you go looking for a spouse through diplomacy, you can check the traits beforehand.
If you use the 'find bride' option in the character menu, you'll be given a potential bride with a random trait. You can then choose whether to marry them or decline the offer. However, you can only receive 1 offer per character per turn, and using this option will cost money each time. So in my opinion it's not a very effective method to get +1 influence.
I just kept all my men single for this achievement.

4. Traits:
Several traits can raise or lower a character's influence as well. However, in most cases it is more or less random which traits a character receives. There is a whole system behind it that you could dive into to make it less random.
But if you are already familiar with that, you probably wouldn't be reading this guide. So I'm going to assume you are like me, and just treat them as randomly assigned things that you'll just have to work with.

Faction bonuses / penalties:
There are also a few faction bonuses that either give + or - influence for a few turns or as long as the effect lasts. Since this usually applies to all characters, this is not really relevant for this guide as the difference between characters will remain the same.

And finally 3 tips / words of warning when going for this achievement:

1) Avoid giving characters more influence than your faction leader as much as possible. When characters have more influence than your faction leader they lose loyalty. I suggest keeping your faction leader's influence the highest, and keep the rest 1 or 2 points behind or with the same influence as your faction leader.

2) Keep in mind that new characters might have a much lower influence. There are several ways in which new characters can (suddenly) be added to your faction:

• a boy comes of age.
  
• a female character marries (except when you marry her off to another faction through diplomacy)
  
• a general dies (and you don't have any statesmen to replace him, you'll be forced to select a new character).

To avoid this last one it is best to keep one of your characters as a statesman, just in case one of your generals dies in battle or between turns.

3) Make plenty of save files (every couple of turns). That way you can easily jump back to a certain point if something goes wrong. I would also suggest writing down the influence difference for every turn. Especially if you're as chaotic as me, and might otherwise forget to keep an eye on influence for several turns (which then makes it hard to know how far to go back).