COUNTIF can accept only one criteria, you seemingly have 2 - date & name... pre XL2007 your options are quite limited, SUMPRODUCT (or CSE Array) or use Concatentaion so as to negate need for SUMPRODUCT etc...

if your data table is relatively small and number of SUMPRODUCTS relatively few then SUMPRODUCT is not a bad option per se, eg:

H5: =SUMPRODUCT(--(Data!$A$3:$A$92=$A5),--(Data!$G$3:$G$92<H$3),--(Data!$G$3:$G$92>=I$3))
copied across to J5 and down for rows as required

NOTE: your data types in column G on DATA should really be consistent, your first entry for Barker. J for ex is a date stored as text, you could negate that with:

H5: =SUMPRODUCT(--(Data!$A$3:$A$92=$A5),--(0+Data!$G$3:$G$92<H$3),--(0+Data!$G$3:$G$92>=I$3))

but if G were to contain non-numeric text strings (like a Null or "apple") the Sumproduct would error.