Step back for a moment, and simply assume R is dedekind. Mod out by the ideal Mj+1 to give a quotient ring S. By correspondence, let H be the image of Mj in S. There is nothing between 0 and H, since that would pull back to something between Mj and Mj+1 in R. This makes H a simple S module, hence it is isomorphic to S mod a maximal ideal. Since M drives Mj into Mj+1, M drives H into 0. Thus H is isomorphic to S mod a maximal ideal containing M. The maximal ideal is M, and H = S/M. By correspondence, this is the same as R/M.
Now H is an R module, and with MH = 0, H is an R/M module, or a K vector space. It is also isomorphic to K. Therefore, H = K. This holds for every quotient Mj/Mj+1.
When M is principal, generated by t, you can map K onto each quotient without resorting to the axiom of choice. The isomorphism between K and the simple R module H is determined by the image of 1, which is arbitrary. (This was discussed in the introduction to simple modules.) Map 1 to tj, and the isomorphism is fixed. Each successive quotient is isomorphic to K, in a canonical fashion.
Now return to R a dvr. Let R′ be the completion of the valuation ring R, and let F′ be the completion of the field F, using the valuation metric. Now R′ is a complete dvr, or a cdvr. The elements of R′ are uniquely represented by power series in K[[t]], and the elements of F′ are laurent series. Addition and multiplication follow the usual polynomial rules, though there may be some carry operations, depending on R. You may want to review the section that describes these representations.
For illustration, take R = Z7 (Z localized about 7), t = 7, M = 7*R, and K = Z7 (Z mod 7). (Sorry about the notational ambiguity.) Take the completion of this dvr to get the 7-adic numbers. Represent the integer 423, which is part of the cdvr R′ and the original ring R. We're really converting to base 7. The resulting sequence is {3,4,1,1,0,0,0…}. This is the reverse of the usual notation for base conversions. The constant term comes first, and the higher powers of 7 flow off to the right. The left right confusion comes from our conventions on polynomials, with the constant term at the right, and power series, with the constant term at the left. The completion of R extends polynomials into power series, and that switches things around. This can be especially confusing when you add to 7-adic numbers together. The carry operation flows to the right, rather than the left. Just take what you learned in elementary school and reflect it.
The cdvr is determined by the residue field K and the carry rules. In the above example, K is Z mod 7, and the arithmetic carry rules apply. Remove these rules, and the cdvr becomes the formal laurent series in t with coefficients in K. Thus 4t + 5t = 2t, rather than 2t+t2. It's a different cdvr with the same residue field.
The carry rules do not change from one digit to the next. If 3+5 = 1+t+2t2, then multiply through by tj, and the same relationship holds at position j in the series. This is valid for multiplication as well. Note that the spillover need not be confined to the next digit, as it is with the p-adic integers; it could ripple all the way down the series. The cdvr is uniquely determined by the field K, and the carry rules for the sums and products within K, i.e. in position 0.
Remember that R could be a pid, with a maximal ideal M. Localize about M to get a dvr. If M is principal, generated by t, then the maximal ideal, after localization, is also generated by t. Complete this to get the cdvr. Notice that K = R/M has not changed, from the pid, to the dvr, to the cdvr.
This is not a universal definition. Some say the reals or rationals, with the usual distance topology, form a local field. Some allow any characteristic, or any residue field.
A uniformizer of a local field is an element with valuation 1, a generator of the maximal ideal.