I haven't actually built live chat into a forum, though I think there is a vBulletin compatible plugin or addon.
Blog/forum integration is more common, and depends on your starting point. If you are built on Wordpress, then you need to find a forum mod to hook them up. Many CMS programs have both, either built in or as addons. Forum software like vBulletin comes with a blog option, though you are likely to find it somewhat less robust than Wordpress.
The obvious problem with live chat is moderation (or lack of it) and access (one on one is not quite as exposed as one to many, many to many). Depends on the type of chat, age of the users, and the location (countries). There are many forum softwares that are available, some in Perl, some in PHP, etc. Most can be folded into exisiting websites and skinned appropriately.
Do you worry that live chat will take attention away from the forum? I have thought about adding a live chat hack to my vBulletin site, but then I think that if they're chatting in the chat box, they're not posting in the forums.
There are many chat modules out there done in java (wrapped in PHP). Although cheap and popular they have issues. Some PC's who's java is not in full health fail to load the app fully or it's crippled. The second issue is the one time java security pop up. If they answer 'no' or 'never' they don't get chat. Some people won't run unsigned java apps. Signing is simple and free to do (with enough searching).
There are a few AJAX based chat modules that have the added bonus of running on low end cell phones. I've made a few with both types on a few sites, I've always settled on the AJAX ones. Search on 'ajax chat client' for a lot of good ones.
Many module authors host their own small IRC network for your chat room. Not having to build a chat server is a BIG plus. Quite a few allow branding and skinning customization of the chat client and more secure chat room modes.