you can share any internet connection in theory, but in reality sharing a dial up connection will mean that both computers access the net at a snails pace.
Do you have a BT phone? if so, sign up for a basic adsl account at www.plus.net
. some accounts are under £15 i think.
what OS do you have. I think from Win 98 SE, sharing a network connection is possible. WIndows XP makes it very easy.
Or, when you sign up for ADSL, buy a adsl ethernet router, then both computers plug directly intot he "internet connection" so you dont have to rely on one computer being on for the other one to access the internet.
MAKE SURE YOU HAVE A FIREWALL and ANTIVIRUS software!!!