First, you have to understand that Rudolf Clausius put together his ideas on entropy in order to account for the losses of energy that was apparent in the practical application of the steam engine. At the time he had no real ability to explain or calculate entropy other than to show how it changed. This is why we are stuck with a lot of theory where we look at deltas, calculus was the only mathematical machinery to develop the theory.
Ludwig Boltzmann was the first to really give entropy a firm foundation beyond simple deltas through the development of statistical mechanics. Essentially he was the first to really understand the concept of a microstate which was a vector in a multidimensional space (e.g. one with potentially infinite dimensions) that encoded all of the position and momentum information of the underlying composite particles. Since the actual information about those particles was unknown, the actual microstate could be one of many potential vectors. Entropy is simply an estimate of the number of possible vectors that actually could encode the information on the particle positions and momentums (remember, each individual vector on it own encodes the information about all the particles). In this sense entropy is a measure of our ignorance (or lack of useful information).
It is this latter use of entropy to measure our level of knowledge that led Claude Shannon to use the machinery of entropy in statistical mechanics to develop information theory. In that framework, entropy is a measure of the possible permutations and combinations a string of letters could take. Understanding information entropy is very critical to understanding the efficacy of various encryption schemes.
As far as defining Temperature in terms of entropy. These are general viewed as being distinct but related measures of the macrostate of a system. Temperature- entropy diagrams are used to understand heat transfer of a system. In statistical mechanics the partition function is used to encode the relationship of temperature and entropy.
Helpful Websites
This website is very useful; see eq 420, temp is embedded in definition of beta. This websiteThis website explains Rudolf Clausius' definition of entropy. This websiteThis website talks about Claude Shannon and his contributions to information theory. This websiteThis website explains the history of entropy and some of the different definitions. This websiteThis website talks about Ludwig Boltzmann's life and definition of entropy. This websiteThis website further explains the relation between temperature and entropy.
