My character starts with a single Barrier at level 1, but getting "The Six" Barrier Abilities requires increasing all five of the other Barriers to level 1 as well. If I min-max a character specifically to get to The Six, what's the minimum amount of Essence I would need to do so? What would this character look like?
You need at least 60 Essence to access the first level of The Six.
There are two optimum character builds for this purpose.
The first option is a zweyjen with their Affinities in Emotion, Physic, and Shadow set to +3, and the other three Affinities set to +1, +0, and -4 in any configuration. The character's starting Barrier should be in whichever Energy gets the -4 Affinity. It will require 10 Essence each to increase their Emotion, Physic, and Shadow Barriers to 1, and then 15 Essence each to increase the other two. That totals 60.
The other option is similar: a human with Emotion Affinity +3, any other two Affinities also set to +3, any one set to -4, and the last two at +0. Again, the character's starting Barrier should be in the Energy with the -4 Affinity. It costs 10 Essence each to open three of the Barriers, and 15 each for the remaining two, totalling 60.
The human is obviously the more versatile choice, since you get to pick which two other Affinities you want (which will influence your starting Talent and Defense stats).
What we're doing here is taking advantage of the quirks of the Affinity stats, described on p. 90 of the Rulebook, which are the only stats that determine the Essence cost of a Barrier increase. Tanking one Affinity to -4 (the minimum allowed) gives us 5 Affinity points on top of the 5 that we get naturally. It would be very expensive to increase our Barrier in the Energy that we're tanking, but choosing that Energy as our starter means we get the Barrier for free.
We spend the 10 Affinity points getting as many Affinities as possible to +3, since that value minimizes the Essence cost of level-ups. For our zweyjen, it costs 3 points each to bring their Emotion, Physic, and Shadow Affinities to +2, then their +1 racial bonus to these stats sets the final value to +3. The last 1 point serves no purpose. For our human, they naturally start with +3 Emotion, and then it costs 5 points each to bring two other Affinities to +3. No other race can be used, since every other race gets at best one +2 or two +1 Affinity bonuses.
A side note: Min-maxing to get to the second level of The Six is a more complicated problem. As a benchmark, though, our two characters above would require an additional 72 Essence to get there: 8 each to raise their three Barriers with +3 Affinity, 24 for the one with -4 Affinity, and 12 each for the last two.