What you're talking about is commonly referred to as "tamping". This is generally only used in making espresso, and is not recommended for any other brew method that I know of.
You are correct in assuming that the tamping will increase brew times and therefore the coffee will be in contact with the water for longer. This of course produces a stronger coffee.
However, this is not the "correct" way to increase the strength of coffee. The generally accepted way of doing this is changing the brew ratio. The brew ratio is the ratio of grounds to water, typically denoted in grams. The ratios range greatly depending on type of brewing. Tamping the coffee will also affect the flavour of the coffee, because one part of water will extract from one part of grounds longer, producing an undesirable taste.
If you want stronger coffee, simply add more grounds to your filter basket. This way your water is not in contact with the grounds for too long, but rather extracts from more grounds. The flavour will be noticeably better through this method.
Hope this helps!