The usual problem with this is that the cable you're using isn't a genuine apple cable but as you've said you have tried the original then the only other explanation is that it is the charging block itself that is the problem. After a while with iPhone 5's lightening charging block is that the gold plates inside start to score and once they are marked they are damaged so they will not function as how it used to do. Only suggestion is that you consider replacing the charging port that you can purchase from here and follow some simple steps also provided by iFixit. Hope this helped.
Part: [product|IF118-003]
Tutorial: [guide|10702]