Coffee Cleaning – What is This Scum in My Coffee?

You do want a coarse grind for french press coffee. The key is letting the coffee steep for long enough to make a strong brew. I have good results with a 5 minute steep time. Be sure to use a timer, and compare different times to find the optimal brew. As a general rule, if you are at altitude, then you need to add time because boiling water is cooler. (Note that I am at 5000ft, so 5 minutes may be too long for you.)

Below is a picture of the coffee chunks in a coarse grind on top of a US 5 cent coin (21.21 mm in diameter).

The picture is from this article.

EDIT: The reason to use a coarse grind for a french press is that finer grinds will get stuck in the filter and/or push their way through. This has three potential problems: 1) it will be harder to press down the filter, possibly leading to spillage or breakage; 2) your coffee will have grinds in it; 3) the coffee may become bitter, as the finer grinds that passed through the filter remain in contact with the coffee for too long.

To answer your question with a straightforward solution, my key recommendation would be to buy a new machine for two reasons: better espresso and milk.

Before all, if you are not willing to invest more than $100 (USD) in a machine (more like $350, but I'm sure you can scrounge for a decent entry-level machine to get the job done), you can stop reading as my suggestion relies on a better machine. That said, quickly browsing Amazon I found a Cuisinart unit with tepid reviews (205 averaging 3.5 stars) on sale for about $80.

Here is what is going wrong with your latte:

• Machine: If you have a Krups it's pressure probably isn't rated. This is a bad sign as it indicates how well it will "pull" (brew, make) the espresso. My guess for most units: Unrated = won't pull. Buy a machine with a rated pressure, 15-bar is what to expect. (I don't mean to sound snobby; I got by with a $60 Coffeemate for awhile, but they won't get you the results you're asking for)
• Bean: Lattes are made with espresso, not coffee (U.S.). You're using coffee. So start by buying espresso beans instead of dark roast coffee or Folgers. While you're at it, if you want it to taste like restaurant quality, buy them from a local roaster (I'd imagine you can get some mad fresh beans in Africa) and grind them fresh (just prior to brewing). I prefer a coarser grind to my espresso, you will also need to press the grinds ("pack") in the gruppa (metal basket of machine).
• Milk: While you're at it, don't necessarily use whatever comes out of the fridge. Here and here have the dirt on milk and frothing. Basically fuller body for the drink use milk with more fat, if not use less fat. If you have a favorite place in mind for your exemplar latte, go ask them where they get their milk and follow suit. There's a degree of preference in which milk you use, go with the one you know you prefer and perfect your technique.
• Frothing/Steaming: You aren't frothing your milk. Buy a machine with a working wand, froth the milk. Actually, frothing the milk is fun and it tastes great all on its own. Note: you do want to get foam on top, this indicates you have properly steamed the milk below (remember to hold back the foam, and pour only the steamed milk into your latte, then depending on your region's definition adding a bit of foam on top).
• Container: Add some dignity to life whenever you can. Use ceramic; you're at home.

Those are the key differences that are causing problems for the latte you want to amp up. Now you can skip the consumer espresso machine advice and get a stove-top setup (espresso pot and use a normal, smallish pot with a whisk for the milk), but that takes a smidge more effort in terms of getting it right. Either way, following these guidelines should get that latte much closer to where you want it to be. Basically, the only advantage the shop will have over yours is a bigger machine, overworked employees who might not have the time to care about your coffee at this time, and paper cups.