Classification of one-dimensional representations, that is, characters $ G_K to E ^ times $, is a little more complicated than you suppose in your question. Such a character lands in $ O_E ^ times $ by compactness; and since $ O_E ^ times $ profinite, and class field theory identifies the abelianization of $ G_K $ with the professional completion of $ K ^ times $, we conclude that there is a canonical bijection between continuous characters $ G_K to O_E ^ times $ and continuous characters $ K ^ times to O_E ^ times $.

With regard to this bijection, we can read which characters are crystalline, semi-stable, of Rham or Hodge-Tate using the restriction of the character to $ O_K ^ times $: Details appear in Appendix B of B. Conrad, "Lifting Global Representations with Local Properties"; see also this question and this question.

As for $ n = $ 2, the best answer I can give is that for $ K = mathbf {Q} _p $, there is a bijection between two-dimensional representations of $ G _ { mathbf {Q} _p} $ and certain $ p $Banach representations in space $ GL_2 ( mathbf {Q} _p) $: it's the $ p $Ado-Langlands local correspondence of Colmez. You can then classify the 2-dimensional representatives that are Rham / crystalline according to their $ GL_2 $ representation. However, it is a bijection between a type of extremely complex object and another type of object also complicated; there is no hope of obtaining a realistic setting of all the representations involved. In addition, the extension of this correspondence to two-dimensional representatives of $ G_K $, for arbitrary K $, or to the three-dimensional representatives of $ G _ { mathbf {Q} _p} $, is an open problem despite a decade or more of very intensive efforts.

(Another point of view: if you just want to classify crystalline / semistable / Rham representatives, but you do not necessarily ask for a classification of *all* reps and how the crys / ss / dR sit inside, you can do it in some cases by sorting all the filters $ varphi $-modules, $ ( varphi, N) $-modules, etc. fulfilling the required conditions. See for example this paper of Dousmanis. But that does not tell you anything about what the non-Rham guys look like.)