First, we need to know what you already know about the user. Obviously, you have a username and an old password. What else do you know? Do you have an email address? Do you have data regarding the user's favorite flower?
Assuming you have a username, password and working email address, you need to add two fields to your user table (assuming it is a database table): a date called new_passwd_expire and a string new_passwd_id.
Assuming you have the user's email address, when someone requests a password reset, you update the user table as follows:
new_passwd_expire = now() + some number of days
new_passwd_id = some random string of characters (see below)
Next, you send an email to the user at that address:
Dear so-and-so
Someone has requested a new password for user account <username> at <your website name>. If you did request this password reset, follow this link:
http://example.com/yourscript.lang?update=<new_password_id>
If that link does not work you can go to http://example.com/yourscript.lang and enter the following into the form: <new_password_id>
If you did not request a password reset, you may ignore this email.
Thanks, yada yada
Now, coding yourscript.lang: This script needs a form. If the var update passed on the URL, the form just asks for the user's username and email address. If update is not passed, it asks for username, email address, and the id code sent in the email. You also ask for a new password (twice of course).
To verify the user's new password, you verify the username, email address, and the id code all match, that the request has not expired, and that the two new passwords match. If successful, you change the user's password to the new password and clear the password reset fields from the user table. Also be sure to log the user out/clear any login related cookies and redirect the user to the login page.
Essentially, the new_passwd_id field is a password that only works on the password reset page.
One potential improvement: you could remove <username> from the email. "Someone has request a password reset for an account at this email address...." Thus making the username something only the user knows if the email is intercepted. I didn't start off that way because if someone is attacking the account, they already know the username. This added obscurity stops man-in-the-middle attacks of opportunity in case someone malicious happens to intercept the email.
As for your questions:
generating the random string: It doesn't need to be extremely random. Any GUID generator or even md5(concat(salt,current_timestamp())) is sufficient, where salt is something on the user record like timestamp account was created. It has to be something the user can't see.
timer: Yes, you need this just to keep your database sane. No more than a week is really necessary but at least 2 days since you never know how long an email delay might last.
IP Address: Since the email could be delayed by days, IP address is only useful for logging, not for validation. If you want to log it, do so, otherwise you don't need it.
Reset Screen: See above.
Hope that covers it. Good luck.