By bus  Source from Wiki travel, http://wikitravel.org/en/Almaty

From Urumuqi从乌鲁木齐

Sleeper buses leave from Urumqi at 6pm every day of the week except Saturday and take approximately 24 hours to arrive in Almaty ( Tickets for a lower bed are Y420, upper beds are Y400, and a berth in the back bed of the bus is Y380. Buses depart from the international branch of the Nianzigou Bus Station (碾子沟客运站), which is located about 50 meters to the left of the main Nianzigou station (if facing the front of the station) on the other side of the Wenshabao (温莎堡) building. Beware that the crossing at Korgas (霍尔果斯) closes on Chinese national holidays (including the first week of October for National Day). 

除周六外， 每天晚上北京时间18：00都有从乌鲁木齐开往阿拉木图的国际长途公共汽车， 旅程约为24小时。 （下铺为420元， 上铺为400元， 在车后方的多人铺为380元）， 在碾子沟客运站国际长途车处发车， 其位于碾子沟客运站右侧50米处温沙堡楼的另一侧。 需注意霍尔果斯口岸将在中国节假日闭关。 （包括国庆节的第一个星期）

From Yining从伊宁

Buses also leave from Yining and it takes about 10 hours to go to Almaty. They cost USD 30 or Y150. These buses leave two or three times in a week, ask the busdrives in Yining when they will go. You could also take a bus to Korgas from Yining (Y30 - Y38) and go to Kazakhstan by foot from Korgas. After being on the other side of the border you could take a taxi which will cost about 3000 Tenge to go to Almaty. The trip from Korgas to Almaty is about 4 to 6 hours. The city of Tacheng city (north of Korgas and north of Yining) also has buses that run several times a week to Almaty and back.

从伊宁也有从发往阿拉木图的国际班车，车程约为10个小时。 车票约为150元， 一周发车两至三次， 具体时间须询问在伊宁的国际班车司机。你也可以从伊宁市坐公共汽车到霍尔果斯（40元）再从霍尔果斯口岸步行过关， 通过对方口岸后可乘坐出租车（3000坚戈一位）前往阿拉木图。 从霍尔果斯到阿拉木图大致需要4到6个小时。 塔城市每周也有几次往返阿拉木图的国际班车。