I’ll assume that user research has convinced you that it’s preferred that users select their country first, then their time zone, as opposed to the user selecting their time zone alone (e.g., users don't know their time zone in GMT+/- format).
There are various ways to graphically tie the two dropdowns together:
Place Time Zone directly below Country so their respective top and bottom abut each other.
Suppress borders of both dropdowns and supply you own tightly wrapping border than surrounds both controls.
Place both controls on a shared background shade or color (easiest to do in a web app). A subtle shade is all you need and shouldn't be too jarring.
As for behavior, disable Time Zone until the user selects Country. When the user selects a country, populate and enable Time Zone and automatically advance focus to the Time Zone for its selection. Ideally, the Time Zone dropdown should open revealing the new list. It’s also probably preferred that you default to the correct time zone for countries with a single zone. If the user changes the country, then repopulate the Time Zone, advance focus, and automatically select either the equivalent or default zone as applicable.
If you have the space, maybe a tree control, rather than two dropdowns, would be a better approach. However, overall this seems like a lot of work for the user –having to choose among ~200 countries in order to determine which of 40-odd possible times zones to use. Your requiring the users to enter more information than you’re actually using, assuming you don’t really care about their countries, just the time.
The simplest alternative (for the user) is to automatically determine the user’s rough geographic location from their IP address and use that to set the local time. If this is prone to error, then provide the user with a means to adjust the time (e.g., spinner buttons beside the clock). You’ll usually be in the ballpark if not exactly right.
Another approach is to ask the user to enter the current local time. Compare it with your server time to get the right time zone (rounding the difference to the nearest quarter hour). Handling daylight savings time can be an issue, but that’s true with your two-dropdown design too. And it may not make much difference if the time is off by an hour, depending on your application (e.g., if it's just to let other users guess when the user is at work or may be asleep).