Wanting to know if you can use a compact crank and also use a compact cassette , I have placed both on my bike but the chain seems to be at the extreme when placing the chain in the smallest gear and then also the biggest gear. Or is this normal?
This is probably normal, but if the chain goes slack in small/small or completely taught in big/big, you may need to adjust the length of your chain accordingly. If it does both, you likely need a longer cage derailleur, but keep reading and you'll know how to check on that last case and know for certain.
Each cage length for a given derailleur is designed to take a minimum sprocket size, a maximum sprocket size, and a maximum number of teeth difference between running the bike in big chainring/big sprocket and small chainring/small sprocket known as "maximum capacity" or "total capacity". That final number basically defines how much extra chain the derailleur cage can take up and thus how wide of a gear range you can run. For example, a Shimano 105 RD-5700 rear derailleur has a minimum sprocket size of 11t, a maximum sprocket size of 28t, and a total capacity of 33t. To find out whether your drivetrain is within the total capacity range of the rear derailleur that you're running, use the following formula:
([teeth on largest chainring] + [teeth on largest cog of cassette]) - (teeth on smallest chainring] - [teeth on smallest cog of cassette])
for a 50/34 crankset in combination with an 11/23 cassette, you would have
( 50 + 23 ) - ( 34 + 11 ) which is equivalent to
50 + 23 - 34 - 11 = 28. You might see different orderings of this formula, but it's simple arithmetic so the number will come out the same.
If the number comes out to less than or equal to your derailleur's total capacity rating then you should be fine.
It's worth noting that you can push this if you feel comfortable doing so, but it's likely that the gears you're supposed to stay out of anyway (the aforementioned big/big, or small/small) will become totally unusable. For this reason it's probably best that you just adhere to the manufacturer's rating.
You need to adjust your chain length. If you just replaced your chain rings and sprockets with smaller ones, that is to be expected. Either visit your local bike shop or do it yourself. See Sheldon Brown's article on the subject of derailer adjustment that also covers chain length.