I've gathered that there was some affinity for/between Alexander II and Lincoln (and they have somewhat similar legacies in their countries). But I've never read anything that explained the reasoning for Alexander's blatant Union boosterism, especially compared to the hedging, neutrality, and outright Confederate sympathizing from other European governments.
I'm not an expert in these matters, and I know there are plenty of users who can provide better background on this, but I know Russia feared a war with Britain, particularly after Russia put down the January Rising in Poland, and they were keen on finding an ally. There was a question on here a while ago, I forget what it was specifically, but one of the answers detailed how Russia actually sent a large portion of its Navy to the US during the war, ostensibly as a show of support, but really to avoid its being uselessly blockaded in St. Petersburg by the Royal Navy in the event of war. I'll try to find the exact thread.
Edit: Found it