From Tbilisi to Baku you can go via train #38/37. It goes every day, so there is no problem with that.
From Baku you can get to Aktau (Kazakhstan) via ferry. People who used this ferry are saying it has no stable schedule, so you can contact ferry station by phone:
(99412) 493-19-63
(99412) 498-10-13
Also you can go to Türkmenbaşy (Turkmenistan), if you want – it much closer, but I can’t find info about transit visa for Turkmenistan.
From there you can pass to one of the main border crossing points Dostyk by train #114 (and even to Ürümqi, China by train #014, since in Almaty there is a Chinese embassy).
Ürümqi is a large rail point in North China, but the only two points to cross the China-Mongolia border is Zamyn-Üüd and Erenhot, so you need get a train to one of these points.