As far as I know the most accessible proof of Bing's construction is in Section 2 of the paper:
J. C. Mayer, L. G. Oversteegen, E, D. Tymchatyn, The Menger curve. Characterization and extension of homeomorphisms of non-locally-separating closed subsets. Dissertationes Math. (Rozprawy Mat.) 252 (1986), 45 pp.
The paper can be downloaded from:
http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1d60f6d5-8102-4734-9186-d4b4bfd8f2dc
The paper does not address construction of a convex metric, but many years ago I wanted to learn the proof of the existence of a convex metric (I never did) and this is what Tymchatyn wrote to me (in 2007):
Once you have appropriate partitions the construction of a convex metric is relatively easy. Bing's 1952 BAMS paper Partitioning Continuous Curves is the place to read it.
There is a proof of the partitioning theorem for Peano continua in the book of Hall and Spencer,Elementary Topology,Wiley,1955. As I recall the proof there is the one given by Bing. The first part of Bing's proof can be simplified a bit using a Peano map of the interval onto a Peano continuum $X$ to get a decomposition of $X$ into finitely many nice bricks between a closed subset $A$ of $X$ and a point $p$ outside $A$.
Bing assigns weights to elements in a defining sequence of partitions $U_i$ of $X$ in such a way that shortest chains in the partitions between two points converge to a distance which gives the convex metric.