I have two computers connected via LAN network , one of them has internet connection , How can I access the internet from the another computer using the connection in the first one ?
What OS are both of the computers running? What type of connection is currently providing Internet access to the one computer?
Having not mentioned OS, I'm going to assume Windows. Also, by connection via "LAN" I'm assuming you mean via ethernet, and having both of their ethernet ports filled I'm assuming that the Internet connected one is probably using a wifi connection.
If this is all correct, then you should do the following on the computer WITH Internet access:
Open the Control Panel and go to Network Connections. Depending on your version of Windows, you may need to follow an additional link named "Network Connections" or "Manage Network Connections".
You should now see individual icons for each network connection type your computer supports (probably two: "Local Area Connection" and "Wireless Network Connection")
You want to right click on the method by which the internet connection is available (wireless) and select "Properties".
Within the "Properties" window, go to the "Advanced" tab.
Here, all you need to do is check the box labeled "Allow other network users to connect through this computer's Internet connection".
If there are more than 2 network connections available, there will be a drop-down list to select on which network connection you want to provide Internet access.
You need to configure the computer to share the connection. This ENTIRELY depends on your OS. Linux can do it through masquerading/NAT with iptables (for example), Windows uses Internet connection sharing, blah blah...without actual details of what you're doing with what OS and what resources you have available, this is an impossible to answer question.
You can setup a Remote Desktop Session and connect to the machine with the working internet. Through that RDP session you can access the internet and do anything else on that machine.
Assuming you have Windows here is a simple RDP tutorial: http://teamtutorials.com/windows-tutorials/windows-remote-desktop-connection#.T0vSefVW18E
If you use another OS there are plenty of other RDP type applications you can use like GoToMeeting and others...