Once you have your amino-modified oligo, don't dilute it with Tris-HCl since it has competing amino residue in it. Use only the oligo dissolved in water for the reaction.
Dissolve the FITC in DRY DMF or DMSO, 10mg dye in 100 uL DMF, dissolve well. if the coupling is not enough, you can increase the amount of dye but dissolve well.
prepare the Borate buffer (0.1M sodium tetraborate) and adjust the pH to 10.5. Dissolve the oligo with this buffer.
Mix the oligo solution and dye solution in one tube, and vortex. place it on shaking platform and vortex one in 10 mins for first 30 mins. Avoid light during the reaction. The reaction almost ends at 4 hrs (reaches the equilibrium).
After the reaction is over, It is preferred to make gel filtration to eliminate the excess dye. We do the reverse phase HPLC to further purify the product to obtain the pure product.
MarkerGene has their Cy3 and TAMRA "OliGlo" kits that label the phosphate groups (terminal and backbone): http://www.markergene.com/Categories/nalabel.php