The question is indeed somewhat too vague, since a knowledge of certain Ext$^1_\chi$ would be enough to prove the Kazhdan-Lusztig Conjecture: take $M_1$ to be a Verma module and $M_2$ to be a simple highest weight module on which U($\mathfrak{g}$) acts with the the same central character $\chi$.    

References include papers by David Vogan in the 1970s which approached the conjecure.  See also my graduate textbook GSM 94 (Amer. Math. Soc., 2008), Chap. 6 and 8, citing early work by P. Delorme and others; lists of errata on AMS page and my homepage.