Direct online ticketing from the Ministry of Railways website will likely not be an option for you since it requires Chinese reading ability and a Chinese-bank issued credit/debit card. You would have to use a more user-friendly third-party site such as www.travelchinaguide.com (there are others as well). They will add a per-ticket commission which may or may not be worth it to you. However, the advance ticket sales only begins 10 days ahead of departure (12 days for direct Railways site), so even a third-party agent will just be taking an order for you, and will have staff fulfill once the sales period for your ticket dates opens up. Your name and passport number will be needed for all ticketing--many agents will want you to scan/email or fax relevant page copies. Agents will not guarantee ticket procurement around major Chinese holiday periods.

The Z trains to Xi'an are popular but the 2-person deluxe soft sleepers, somewhat less quick to sell out since they are expensive (on a par with flying). So if you will be in Beijing for several days to a week before traveling to Xi'an, it may be possible to just wait and deal with purchase once you arrive--with backup plan such as alternate train, regular soft sleeper, or flying as your Plan B in case of sellout. June is not a particularly heavy travel month in China.