Since you have a 648 and not the later 652/658 model, you don't have to worry about which version of the power supply you have, so HerronScott's 648 list
and its derivatives are what you can safely go by.
Basically you replace every cap except the big 200 Volt one and the little bitty ones.
Go ahead and do all the ones he lists, one at a time and keep notes, and that'll save you going back in a few months when more of the original ones go bad.
The white stuff is glue that holds them in place while they get batch soldered during manufacturing.
You can probably bust it loose with pliers.
Which reminds me, you'll need something to clip off the extra lead length on the replacement caps after you solder them in.
That Elenco set to which you linked doesn't specify the Wattage of the iron.
One of the reviews says it's 30W, another says it's 25, I says it's not quite enough for this particular job because for the negative lead of the capacitors you need something that can put more heat into a solder joint that's part of a large copper ground plane which will be radiating the heat away as you are applying it. That little thing won't stand a chance.
Somewhere in at least one of threads around here I provide a link to the Radio Shack page for a de-soldering iron which they sell. It looks like the unplanned child of a soldering iron and a small turkey baster.
It's one of the few things I'd recommend from Radio Shack.
Since it heats up the solder joint enough to remove the solder with its vacuum bulb, it'll heat up the joint enough to melt solder when installing the replacements.
Here's pretty much the same thing under a different brand name and available from Amazon
Although if you've got plenty of spare money, a Weller 100/140 Watt soldering gun will easily do the job and last for years.
As long as you aren't doing this all day every day in poor ventilation, you should probably avoid the extra aggravation and higher melting temperature of lead-free solder.
Get some rosin core 60/40 or 63/37 lead-tin alloy stuff.
One of those plastic tubes of it should be plenty.
Remember that these electrolytic capacitors you'll be installing are polarized. In this particular instance that means the negative lead will go in the hole that's connected to the power supply's common ground.
If you put one in "backwards", nothing good will happen and it'll probably die very loudly when you plug the TiVo in.
The silkscreening on the power supply board where the caps go should be a circle around the 2 holes, with slanted lines over the half where the hole for the negative lead goes.
On the caps themselves, if there isn't a string of + marks down one side to indicate the positive lead, then there will be a stripe or line of - marks to indicate the negative lead.