A. Markov's papers?

Obtaining old issues of this journal is difficult, though perhaps some American libraries have it, and you can try Interlibrary Loan (ILL). Alternatively, there are two volumes of collected papers Markov's Collected papers, MR0050525 the papers you are looking for must be in this volume. Our library has this, so you can surely obtain this via ILL.

BTW, historians say that Markov chains were introduced for the first time in his paper of 1913, here is an account available online:

http://www.americanscientist.org/libraries/documents/201321152149545-2013-03Hayes.pdf

It gives a reference to an English translation which must be easily available. Also: http://www.alpha60.de/research/markov/ which has links to some translations.

Here is one of the papers you are asking: http://www.alpha60.de/research/markov/DavidLink_OnARemarkableCase_MarkovTrans_2007.pdf

There is a volume of collected works of Markov published in 1951 (in Russian), which is available online:

http://pyrkov.professorjournal.ru/c/document_library/get_file?uuid=dee29674-473c-4c73-a9f2-cdf58abb182b&groupId=996446

EDIT The claim that "Markov chains were introduced for the first time in his paper of 1913" is completely misleading (as well as several other claims made by Hayes). Moreover, in my opinion, his popular "essay" does not qualify as a scholarly publication at all (to begin with, as he himself admits, he was not able to access Markov's originals - as he does not read Russian - and is based entirely on secondary sources significantly embellished by him).

In addition to the aforementioned volume of Markov's collected works published in 1951 the originals of almost all Markov papers are available online including the very first paper on the subject, Распространеніе закона большихъ чиселъ на величины, зависящія другъ отъ друга (although published in the 1906 volume, the date at the end is indeed March 25, 1907, which just means that the 1906 volume was physically printed only in 1907 - this is pretty common). Most other originals are accessible on Markov's page: Изслѣдованiе замѣчательнаго случая зависимыхъ испытанiй (1907), О связанныхъ величинахъ, не образующихъ настоящей цѣпи (1911), Объ одномъ случаѣ испытанiй, связанныхъ въ сложную цѣпь (1911), Объ испытанiяхъ связанныхъ въ цѣпь не наблюдаемыми событiями (1912), and, finally , the "Onegin paper" Примѣръ статистическаго изслѣдованiя надъ текстомъ “Евгенiя Онѣгина”, иллюстрирующiй связь испытанiй въ цѣпь (1913). The original of the 1908 paper Распространеніе предѣльныхъ теоремъ исчисленія вѣроятностей на сумму величинъ связанныхъ въ цепь is available at the link just provided by yannis.

It is in this latter paper that he introduces what we now know as a Markov chain in a very clear and explicit form (not really different from what one can read in modern textbooks), by saying that it is determined by a transition matrix and an initial distribution (p. 2).

Concerning the history of the earlier stages of the theory of Markov chains, the most comprehensive source is, in my opinion, Souvenirs de Bologne by Bernard Bru (2003). To make it short.

The work of Markov was not as unaccessible as one might think. Translations of his papers were published in French in Acta Mathematica (1910) and in German in the 1912 edition of his probability course. However, it did not help much. Poincaré, who in Chapitre XVI ("questions diverses") of the second edition of his Calcul des Probabilités (published also in 1912) introduced random walks on finite groups for studying card shuffling, was not aware of it. As Fréchet and Hadamard put it in their 1933 Sur les probabilités discontinues des événements "en chaîne",

On peut considérer comme une preuve de l’importance de l’étude des probabilités des événements "en chaîne", le fait qu’un certain nombre de mathématiciens s’y sont trouvés conduits indépendamment sans connaître les travaux de leurs prédécesseurs. En particulier, comme Poincaré, comme M.M. Urban, Paul Lévy, Hostinsky, Chapman, comme nous, M. von Mises s’est engagé dans cette étude sans avoir eu connaissance des travaux de Markoff, travaux fondamentaux, mais publiés en russe.

For the first time the work of Markov was acknowledged in the West by Hadamard. In a footnote to his 1928 ICM talk Sur le battage des cartes et ses relations avec la mécanique statistique inspired by Poincaré's 1912 work and published in 1931 (pp.133-139) he very clearly describes how he learned about Markov's work:

Comme me l'a fait remarquer M. Polyà, la question des "grandeurs enchaînées" traitée par Markoff (Ac. Russe des Sciences 1908 et Wahrscheinlichkeitsrechnung, 1912, Anhang II) puis, plus récemment par M. Serge Bernstein (Math. Ann., t. 97), est voisine de celle du texte, qu'elle combine en quelque sorte avec celle de la loi des grands nombres. On y est conduit en faisant correspondre à chaque substitution $S$ une valeur d'une grandeur $x$ et considérant la distribution en probabilité de la somme des valeurs prises par $x$ dans les $n$ premiers coups.

It is notable that Hadamard does not even mention the 1913 Onegin paper :)