Typically, your best bet is to look up the specific area or mountain that you plan to hike. Most states or mountains will have organizations meant to dispatch information about the weather for their specific area of coverage.
In the United States, the National Weather Service often will provide forecasts for specific mountain regions. You should look up the area you wish to travel to with the service to find a forecast. The accuracy depends on how well traveled the mountains are. For example, there are location-specific forecasts for many mountains in Seattle like the Olympic Mountains, the North Cascades, and the Central Cascades because they are so well traveled. These forecasts are routinely updated. Other states will have these as well. While the webpage for Vermont and Northern New York's mountains is not nearly as well-developed, they still boast a solid forecast, though it is harder to read in its all caps format. More in the middle of the spectrum, mountains like the Sawtooth in Idaho have reasonably good websites through the National Weather Services, complete with 7-day forecasts and radar maps. In San Diego, the Riverside County Mountains have a similar webpage.
Outside of the United States, things may get a little bit trickier as not everywhere will have forecasts for you online (or in English)! However, some places are better than others, some even better than the United States. For example, if you were planning on hiking in Scotland, the Mountain Weather Information Service, or MWIS, has a webpage that produces eight forecasts, seven days a week, released at 4:30 PM each day. Hikers can check at the website before they go and they also have a mobile phone application for looking up on the go. Forecasted areas include the Northwest Highlands, the West Highlands, the Cairngorms National Park and Monadhliath, the Southeastern Highlands, the Southern Uplands, the Lake District, the Snowdonia National Park, and the Peak District and Yorkshire Dales.
More generally though, a good website to look at is called Mountain Forecast. The website allows you to look by typing in a mountain or to navigate by range, sub-range, and mountain type. However, though it boasts many mountains, it still lacks smaller ones and does not generate its own reports, relying on nearby weather stations that can still be far away and downhill. For mountains outside of the US though, this can still prove very useful, particularly for more well trafficked mountains.
Whatever these websites say though, remember that forecasts can be incorrect. Always plan ahead and rely on your best asset - your instincts. When you are climbing, there is nothing more important than staying alert.