In aqueous solutions amino acids are usually considered as zwitterions.
Even as solid salt they are charged.
I can't really say, why usually it is drawn uncharged. It may confuse but it isn't important when you want to know which charge it really has. Just take a look at the pH and the pka values.
In a solution the charge depends on the pH of the solution. If there are additional groups which can protonate or deprotonate you have to take a look at their pka.
Since Lysin has to amino groups and one carboxylic acid group.
The pka of the both aminogroups is much higher than your pH of 5, so both are protonated. You have two positive charges.
The carboxylic acid is deprotonated at pH 5 because its pka is 2.18. So one negative charge here.
That means Lysin has in total one positive charges at pH 5.
In water you have always charges because you have a acid and base in one molecule, so they can react with each other. COOH protonates the $NH_2$ and $NH_2$ deprotonates the COOH.
The charge of Isoleucin is an easy example because you have no side chains, just COOH and $NH_2$. You can take a given pH (maybe 5) and then take a look at the pka of both functional groups as written above. Then add the charges.
Or you can take a look at the pI. This works even with side chains. If the pH = pI you have a zwitterion with on positive and one negative charge, so the whole molecule is neutral. If your pH is higher than the pI it is mor basic and something will deprotonate so you have a negative charge.
If pH < pI something is protonated und you have a positive charge.
pI refers to that point there half of all
Just take a close look at the functional groups of the side chains. Since most times pka is refered to aqueous solutions only those groups are given a pka value which can be protonated or deprotonated in water. A Methyl group for example $R-CH_3$ can't be deprotoned in water. so why should they give a pka value? It wouldn't be helpful.