Count number of occurences of particular numbers in list

BinCounts[napajecky, {0,85,5}]
(*    {5, 7, 8, 12, 3, 6, 7, 2, 1, 0, 1, 0, 0, 0, 0, 0, 0}    *)

Notice that the range goes to 85 so that the last bin is $[80,85)$ (from 80 inclusive to 85 exclusive).

Alternatively you could bin directly, without rounding first:

napajecky = {0.227, 0.155, 0.184, 0.178, 0.248, 0.106, 0.295, 0.286,
             0.126, 0.109, 0.172, 0.364, 0.131, 0.268, 0.3, 0.172, 0.324,
             0.359, 0.52, 0.277, 0.169, 0.303, 0.169, 0.097, 0.265, 0.063, 
             0.262, 0.19, 0.095, 0.059, 0.035, 0.051, 0.163, 0.099, 0.038,
             0.093, 0.16, 0.1454, 0.3803, 0.2437, 0.0359, 0.0959, 0.0346,
             0.136, 0.275, 0.003, 0.004, 0.003, 0.003, 0.005, 0.1686, 0.1031};

BinCounts[napajecky, {-0.025, 0.825, 0.05}]
(*    {5, 7, 8, 12, 3, 6, 7, 2, 1, 0, 1, 0, 0, 0, 0, 0, 0}    *)

res = Lookup[Counts[napajecky], Range[5, 80, 5], 0]

{7, 8, 12, 3, 6, 7, 2, 1, 0, 1, 0, 0, 0, 0, 0, 0}

For completeness' sake, Tally:

Range[5, 80, 5] /. Append[Rule @@@ Tally[napajecky], _Integer -> 0]

{7, 8, 12, 3, 6, 7, 2, 1, 0, 1, 0, 0, 0, 0, 0, 0}